From 96b49a4153200f31114d924ee49e4b58a904cc67 Mon Sep 17 00:00:00 2001
From: Michael Schneeberger <michael.schneeberger@fhnw.ch>
Date: Tue, 27 Feb 2024 08:16:57 +0100
Subject: use  or  instead of

---
 polymatrix/expression/init.py                      |  2 +-
 polymatrix/expression/mixins/additionexprmixin.py  | 84 ++++----------------
 polymatrix/expression/mixins/cacheexprmixin.py     |  6 +-
 .../expression/mixins/combinationsexprmixin.py     | 41 ++--------
 polymatrix/expression/mixins/degreeexprmixin.py    | 14 ++--
 .../expression/mixins/derivativeexprmixin.py       |  6 +-
 .../expression/mixins/determinantexprmixin.py      | 38 ++++-----
 .../expression/mixins/divergenceexprmixin.py       |  3 +-
 polymatrix/expression/mixins/divisionexprmixin.py  |  8 +-
 polymatrix/expression/mixins/elemmultexprmixin.py  | 92 +++-------------------
 polymatrix/expression/mixins/evalexprmixin.py      | 26 +++---
 polymatrix/expression/mixins/filterexprmixin.py    | 10 +--
 .../expression/mixins/filterlinearpartexprmixin.py | 19 +++--
 .../mixins/fromsymmetricmatrixexprmixin.py         |  6 +-
 polymatrix/expression/mixins/fromtermsexprmixin.py |  4 +-
 polymatrix/expression/mixins/fromtupleexprmixin.py |  2 +-
 .../mixins/halfnewtonpolytopeexprmixin.py          |  2 +-
 .../expression/mixins/legendreseriesmixin.py       | 14 ++--
 polymatrix/expression/mixins/linearinexprmixin.py  |  2 +-
 .../expression/mixins/linearmatrixinexprmixin.py   |  2 +-
 .../expression/mixins/linearmonomialsexprmixin.py  |  6 +-
 .../expression/mixins/matrixmultexprmixin.py       | 20 ++---
 polymatrix/expression/mixins/maxexprmixin.py       |  6 +-
 .../expression/mixins/parametrizeexprmixin.py      |  6 +-
 .../mixins/parametrizematrixexprmixin.py           |  6 +-
 polymatrix/expression/mixins/productexprmixin.py   | 62 +++------------
 .../expression/mixins/quadraticinexprmixin.py      |  2 +-
 .../mixins/quadraticmonomialsexprmixin.py          |  6 +-
 polymatrix/expression/mixins/squeezeexprmixin.py   | 12 +--
 .../expression/mixins/substituteexprmixin.py       | 26 +++---
 .../mixins/subtractmonomialsexprmixin.py           |  4 +-
 polymatrix/expression/mixins/sumexprmixin.py       | 12 +--
 .../expression/mixins/toconstantexprmixin.py       |  2 +-
 .../expression/mixins/toquadraticexprmixin.py      | 46 +++--------
 .../expression/mixins/tosortedvariablesmixin.py    |  2 +-
 .../mixins/tosymmetricmatrixexprmixin.py           |  8 +-
 polymatrix/expression/mixins/truncateexprmixin.py  | 10 +--
 polymatrix/expression/to.py                        |  4 +-
 polymatrix/expression/utils/broadcastpolymatrix.py | 34 ++++++++
 polymatrix/polymatrix/impl.py                      |  4 +-
 polymatrix/polymatrix/init.py                      | 17 +++-
 polymatrix/polymatrix/mixins.py                    | 12 +--
 42 files changed, 255 insertions(+), 433 deletions(-)
 create mode 100644 polymatrix/expression/utils/broadcastpolymatrix.py

diff --git a/polymatrix/expression/init.py b/polymatrix/expression/init.py
index 06fe23c..f8092d2 100644
--- a/polymatrix/expression/init.py
+++ b/polymatrix/expression/init.py
@@ -226,7 +226,7 @@ def init_from_terms_expr(
 
     if isinstance(terms, PolyMatrixMixin):
         shape = terms.shape
-        gen_terms = terms.gen_terms()
+        gen_terms = terms.gen_data()
 
     else:
         assert shape is not None
diff --git a/polymatrix/expression/mixins/additionexprmixin.py b/polymatrix/expression/mixins/additionexprmixin.py
index 7dbf1d2..e3580f0 100644
--- a/polymatrix/expression/mixins/additionexprmixin.py
+++ b/polymatrix/expression/mixins/additionexprmixin.py
@@ -1,9 +1,8 @@
 import abc
 import math
-from polymatrix.polymatrix.impl import BroadcastPolyMatrixImpl
+from polymatrix.expression.utils.broadcastpolymatrix import broadcast_poly_matrix
 
 from polymatrix.utils.getstacklines import FrameSummary
-from polymatrix.utils.tooperatorexception import to_operator_exception
 from polymatrix.polymatrix.init import init_poly_matrix
 from polymatrix.polymatrix.abc import PolyMatrix
 from polymatrix.expressionstate.abc import ExpressionState
@@ -26,31 +25,6 @@ class AdditionExprMixin(ExpressionBaseMixin):
     def stack(self) -> tuple[FrameSummary]:
         ...
 
-    @staticmethod
-    def broadcast(left: PolyMatrix, right: PolyMatrix, stack: tuple[FrameSummary]):
-        # broadcast left
-        if left.shape == (1, 1) and right.shape != (1, 1):
-            left = BroadcastPolyMatrixImpl(
-                polynomial=left.get_poly(0, 0),
-                shape=right.shape,
-            )
-
-        # broadcast right
-        elif left.shape != (1, 1) and right.shape == (1, 1):
-            right = BroadcastPolyMatrixImpl(
-                polynomial=right.get_poly(0, 0),
-                shape=left.shape,
-            )
-
-        else:
-            if not (left.shape == right.shape):
-                raise AssertionError(to_operator_exception(
-                    message=f'{left.shape} != {right.shape}',
-                    stack=stack,
-                ))
-
-        return left, right
-
     # overwrites the abstract method of `ExpressionBaseMixin`
     def apply(
         self, 
@@ -59,44 +33,14 @@ class AdditionExprMixin(ExpressionBaseMixin):
         state, left = self.left.apply(state=state)
         state, right = self.right.apply(state=state)
 
-        # if left.shape == (1, 1):
-        #     left, right = right, left
-
-        # if left.shape != (1, 1) and right.shape == (1, 1):
-
-        #     # @dataclassabc.dataclassabc(frozen=True)
-        #     # class BroadCastedPolyMatrix(PolyMatrixMixin):
-        #     #     underlying_monomials: tuple[tuple[int], float]
-        #     #     shape: tuple[int, int]
-
-        #     #     def get_poly(self, row: int, col: int) -> typing.Optional[dict[tuple[int, ...], float]]:
-        #     #         return self.underlying_monomials
-
-
-        #     right = BroadcastPolyMatrixImpl(
-        #         polynomial=right.get_poly(0, 0),
-        #         shape=left.shape,
-        #     )
-
-        #     # all_underlying = (left, broadcasted_right)
-
-        # else:
-        #     if not (left.shape == right.shape):
-        #         raise AssertionError(to_operator_exception(
-        #             message=f'{left.shape} != {right.shape}',
-        #             stack=self.stack,
-        #         ))
-
-        #     # all_underlying = (left, right)
-
-        left, right = self.broadcast(left, right, self.stack)
+        left, right = broadcast_poly_matrix(left, right, self.stack)
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(left.shape[0]):
             for col in range(left.shape[1]):
 
-                terms_row_col = {}
+                poly_data = {}
 
                 for underlying in (left, right):
 
@@ -104,26 +48,26 @@ class AdditionExprMixin(ExpressionBaseMixin):
                     if polynomial is None:
                         continue
 
-                    if len(terms_row_col) == 0:
-                        terms_row_col = dict(polynomial)
+                    if len(poly_data) == 0:
+                        poly_data = dict(polynomial)
 
                     else:
                         for monomial, value in polynomial.items():
                             
-                            if monomial not in terms_row_col:
-                                terms_row_col[monomial] = value
+                            if monomial not in poly_data:
+                                poly_data[monomial] = value
 
                             else:
-                                terms_row_col[monomial] += value
+                                poly_data[monomial] += value
 
-                                if math.isclose(terms_row_col[monomial], 0):
-                                    del terms_row_col[monomial]
+                                if math.isclose(poly_data[monomial], 0):
+                                    del poly_data[monomial]
 
-                if 0 < len(terms_row_col):
-                    terms[row, col] = terms_row_col
+                if 0 < len(poly_data):
+                    poly_matrix_data[row, col] = poly_data
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=left.shape,
         )
 
diff --git a/polymatrix/expression/mixins/cacheexprmixin.py b/polymatrix/expression/mixins/cacheexprmixin.py
index e5aef11..22e5a6a 100644
--- a/polymatrix/expression/mixins/cacheexprmixin.py
+++ b/polymatrix/expression/mixins/cacheexprmixin.py
@@ -29,12 +29,12 @@ class CacheExprMixin(ExpressionBaseMixin):
         state, underlying = self.underlying.apply(state)
 
         if isinstance(underlying, PolyMatrixAsDictMixin):
-            cached_terms = underlying.terms
+            cached_data = underlying.data
         else:
-            cached_terms = dict(underlying.gen_terms())
+            cached_data = dict(underlying.gen_data())
 
         poly_matrix = init_poly_matrix(
-            terms=cached_terms,
+            data=cached_data,
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/combinationsexprmixin.py b/polymatrix/expression/mixins/combinationsexprmixin.py
index 9da69af..632efd5 100644
--- a/polymatrix/expression/mixins/combinationsexprmixin.py
+++ b/polymatrix/expression/mixins/combinationsexprmixin.py
@@ -31,19 +31,6 @@ class CombinationsExprMixin(ExpressionBaseMixin):
         self, 
         state: ExpressionState,
     ) -> tuple[ExpressionState, PolyMatrix]:
-        # if self.degree == 0:
-        #     terms = {(0, 0): {tuple(): 1.0}}
-
-        #     poly_matrix = init_poly_matrix(
-        #         terms=terms,
-        #         shape=(1, 1),
-        #     )
-
-        # elif self.degree == 1:
-        #     state, monomials = self.expression.apply(state=state) 
-        #     poly_matrix = monomials
-
-        # else:
 
         state, poly_matrix = self.expression.apply(state=state)
 
@@ -55,13 +42,13 @@ class CombinationsExprMixin(ExpressionBaseMixin):
 
         indices = tuple(gen_indices())
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row, indexing in enumerate(indices):
 
             # x.combinations((0, 1, 2)) produces [1, x, x**2]
             if len(indexing) == 0:
-                terms[row, 0] = {tuple(): 1.0}
+                poly_matrix_data[row, 0] = {tuple(): 1.0}
                 continue
 
             def acc_product(left, row):
@@ -80,29 +67,11 @@ class CombinationsExprMixin(ExpressionBaseMixin):
                 initial={},
             )
 
-            terms[row, 0] = polynomial 
+            poly_matrix_data[row, 0] = polynomial 
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
-            shape=(len(terms), 1),
+            data=poly_matrix_data,
+            shape=(len(poly_matrix_data), 1),
         )
 
-        # indices = filter(lambda v: sum(v) <= self.degree, itertools.product(*(range(self.degree) for _ in range(dim))))
-
-        # state, monomials = get_monomial_indices(state, self.monomials)
-
-        # combinations = tuple(itertools.combinations_with_replacement(monomials, self.number))
-
-        # terms = {}
-
-        # for row, combination in enumerate(combinations):
-        #     combination_monomial = merge_monomial_indices(combination)
-
-        #     terms[row, 0] = {combination_monomial: 1.0}
-
-        # poly_matrix = init_poly_matrix(
-        #     terms=terms,
-        #     shape=(math.comb(len(monomials) + self.number - 1, self.number), 1),
-        # )
-
         return state, poly_matrix   
diff --git a/polymatrix/expression/mixins/degreeexprmixin.py b/polymatrix/expression/mixins/degreeexprmixin.py
index 273add2..71b936a 100644
--- a/polymatrix/expression/mixins/degreeexprmixin.py
+++ b/polymatrix/expression/mixins/degreeexprmixin.py
@@ -26,26 +26,24 @@ class DegreeExprMixin(ExpressionBaseMixin):
     ) -> tuple[ExpressionState, PolyMatrix]:
         state, underlying = self.underlying.apply(state=state)
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
             for col in range(underlying.shape[1]):
 
-                underlying_terms = underlying.get_poly(row, col)
+                polynomial = underlying.get_poly(row, col)
 
-                if underlying_terms is None or len(underlying_terms) == 0:
+                if polynomial is None or len(polynomial) == 0:
                     continue
 
                 def gen_degrees():
-                    for monomial, _ in underlying_terms.items():
+                    for monomial, _ in polynomial.items():
                         yield sum(count for _, count in monomial)
                     
-                # degrees = tuple(gen_degrees())
-
-                terms[row, col] = {tuple(): max(gen_degrees())}
+                poly_matrix_data[row, col] = {tuple(): max(gen_degrees())}
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/derivativeexprmixin.py b/polymatrix/expression/mixins/derivativeexprmixin.py
index 38ea9cc..0b6ec6e 100644
--- a/polymatrix/expression/mixins/derivativeexprmixin.py
+++ b/polymatrix/expression/mixins/derivativeexprmixin.py
@@ -56,7 +56,7 @@ class DerivativeExprMixin(ExpressionBaseMixin):
                 stack=self.stack,
             ))
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
 
@@ -76,10 +76,10 @@ class DerivativeExprMixin(ExpressionBaseMixin):
                 )
 
                 if 0 < len(derivation):
-                    terms[row, col] = derivation
+                    poly_matrix_data[row, col] = derivation
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(underlying.shape[0], len(diff_wrt_variables)),
         )
 
diff --git a/polymatrix/expression/mixins/determinantexprmixin.py b/polymatrix/expression/mixins/determinantexprmixin.py
index 7614669..d848bd2 100644
--- a/polymatrix/expression/mixins/determinantexprmixin.py
+++ b/polymatrix/expression/mixins/determinantexprmixin.py
@@ -26,7 +26,7 @@ class DeterminantExprMixin(ExpressionBaseMixin):
         self, 
         state: ExpressionState,
     ) -> tuple[ExpressionState, PolyMatrix]:
-        raise Exception('not implemented')
+        # raise Exception('not implemented')
 
         if self in state.cache:
             return state, state.cache[self]
@@ -35,7 +35,7 @@ class DeterminantExprMixin(ExpressionBaseMixin):
 
         assert underlying.shape[0] == underlying.shape[1]
 
-        inequality_terms = {}
+        inequality_data = {}
         auxillary_equations = {}
 
         index_start = state.n_param
@@ -43,69 +43,69 @@ class DeterminantExprMixin(ExpressionBaseMixin):
 
         for row in range(underlying.shape[0]):
 
-            current_inequality_terms = collections.defaultdict(float)
+            polynomial = collections.defaultdict(float)
 
             # f in f-v^T@x-r^2
             # terms = underlying.get_poly(row, row)
             try:
-                underlying_terms = underlying.get_poly(row, row)
+                underlying_poly = underlying.get_poly(row, row)
             except KeyError:
                 pass
             else:
-                for monomial, value in underlying_terms.items():
-                    current_inequality_terms[monomial] += value
+                for monomial, value in underlying_poly.items():
+                    polynomial[monomial] += value
             
             for inner_row in range(row):
 
                 # -v^T@x in f-v^T@x-r^2
                 # terms = underlying.get_poly(row, inner_row)
                 try:
-                    underlying_terms = underlying.get_poly(row, inner_row)
+                    underlying_poly = underlying.get_poly(row, inner_row)
                 except KeyError:
                     pass
                 else:
-                    for monomial, value in underlying_terms.items():
+                    for monomial, value in underlying_poly.items():
                         new_monomial = monomial + (index_start + rel_index + inner_row,)
-                        current_inequality_terms[new_monomial] -= value
+                        polynomial[new_monomial] -= value
 
                 # auxillary terms
                 # ---------------
 
-                auxillary_term = collections.defaultdict(float)
+                auxillary_polynomial = collections.defaultdict(float)
 
                 for inner_col in range(row):
 
                     # P@x in P@x-v
                     key = tuple(reversed(sorted((inner_row, inner_col))))
                     try:
-                        underlying_terms = underlying.get_poly(*key)
+                        underlying_poly = underlying.get_poly(*key)
                     except KeyError:
                         pass
                     else:
-                        for monomial, value in underlying_terms.items():
+                        for monomial, value in underlying_poly.items():
                             new_monomial = monomial + (index_start + rel_index + inner_col,)
-                            auxillary_term[new_monomial] += value
+                            auxillary_polynomial[new_monomial] += value
                 
                 # -v in P@x-v
                 try:
-                    underlying_terms = underlying.get_poly(row, inner_row)
+                    underlying_poly = underlying.get_poly(row, inner_row)
                 except KeyError:
                     pass
                 else:
-                    for monomial, value in underlying_terms.items():
-                        auxillary_term[monomial] -= value
+                    for monomial, value in underlying_poly.items():
+                        auxillary_polynomial[monomial] -= value
 
                 x_variable = index_start + rel_index + inner_row
                 assert x_variable not in state.auxillary_equations
-                auxillary_equations[x_variable] = dict(auxillary_term)
+                auxillary_equations[x_variable] = dict(auxillary_polynomial)
 
             rel_index += row
-            inequality_terms[row, 0] = dict(current_inequality_terms)
+            inequality_data[row, 0] = dict(polynomial)
 
         state = state.register(rel_index)
 
         poly_matrix = init_poly_matrix(
-            terms=inequality_terms,
+            data=inequality_data,
             shape=(underlying.shape[0], 1),
         )
 
diff --git a/polymatrix/expression/mixins/divergenceexprmixin.py b/polymatrix/expression/mixins/divergenceexprmixin.py
index 10db5b4..fc12abd 100644
--- a/polymatrix/expression/mixins/divergenceexprmixin.py
+++ b/polymatrix/expression/mixins/divergenceexprmixin.py
@@ -55,8 +55,7 @@ class DivergenceExprMixin(ExpressionBaseMixin):
                 monomial_terms[monomial] += value
 
         poly_matrix = init_poly_matrix(
-            # terms={(0, 0): dict(monomial_terms)},
-            terms={(0, 0): monomial_terms},
+            data={(0, 0): monomial_terms},
             shape=(1, 1),
         )
 
diff --git a/polymatrix/expression/mixins/divisionexprmixin.py b/polymatrix/expression/mixins/divisionexprmixin.py
index 0f2fada..65e6bc3 100644
--- a/polymatrix/expression/mixins/divisionexprmixin.py
+++ b/polymatrix/expression/mixins/divisionexprmixin.py
@@ -55,7 +55,7 @@ class DivisionExprMixin(ExpressionBaseMixin):
                 state=state, 
                 left=left, 
                 right=init_poly_matrix(
-                    terms=right_inv,
+                    data=right_inv,
                     shape=(1, 1),
                 )
             )
@@ -64,7 +64,7 @@ class DivisionExprMixin(ExpressionBaseMixin):
         if self in state.cache:
             return state, state.cache[self]
 
-        terms = {}
+        poly_matrix_data = {}
 
         division_variable = state.n_param
         state = state.register(n_param=1)
@@ -80,7 +80,7 @@ class DivisionExprMixin(ExpressionBaseMixin):
                     for monomial, value in underlying_terms.items():
                         yield monomial + ((division_variable, 1),), value
 
-                terms[row, col] = dict(gen_monomial_terms())
+                poly_matrix_data[row, col] = dict(gen_monomial_terms())
 
         def gen_auxillary_terms():
             for monomial, value in right_poly.items():
@@ -94,7 +94,7 @@ class DivisionExprMixin(ExpressionBaseMixin):
         auxillary_terms[tuple()] -= 1
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=left.shape,
         )
 
diff --git a/polymatrix/expression/mixins/elemmultexprmixin.py b/polymatrix/expression/mixins/elemmultexprmixin.py
index 17be163..bfe6397 100644
--- a/polymatrix/expression/mixins/elemmultexprmixin.py
+++ b/polymatrix/expression/mixins/elemmultexprmixin.py
@@ -23,11 +23,6 @@ class ElemMultExprMixin(ExpressionBaseMixin):
     def right(self) -> ExpressionBaseMixin:
         ...
 
-    # # overwrites the abstract method of `ExpressionBaseMixin`
-    # @property
-    # def shape(self) -> tuple[int, int]:
-    #     return self.left.shape
-
     @staticmethod
     def elem_mult(
         state: ExpressionState,
@@ -53,23 +48,23 @@ class ElemMultExprMixin(ExpressionBaseMixin):
                 shape=left.shape,
             )
 
-        terms = {}
+        poly_matrix_data = {}
 
         for poly_row in range(left.shape[0]):
             for poly_col in range(left.shape[1]):
 
-                terms_row_col = {}
+                polynomial = {}
 
-                left_terms = left.get_poly(poly_row, poly_col)
-                if left_terms is None:
+                left_polynomial = left.get_poly(poly_row, poly_col)
+                if left_polynomial is None:
                     continue
 
-                right_terms = right.get_poly(poly_row, poly_col)
-                if right_terms is None:
+                right_polynomial = right.get_poly(poly_row, poly_col)
+                if right_polynomial is None:
                     continue
 
                 for (left_monomial, left_value), (right_monomial, right_value) \
-                        in itertools.product(left_terms.items(), right_terms.items()):
+                        in itertools.product(left_polynomial.items(), right_polynomial.items()):
 
                     value = left_value * right_value
 
@@ -80,16 +75,16 @@ class ElemMultExprMixin(ExpressionBaseMixin):
 
                     new_monomial = merge_monomial_indices((left_monomial, right_monomial))
 
-                    if new_monomial not in terms_row_col:
-                        terms_row_col[new_monomial] = 0
+                    if new_monomial not in polynomial:
+                        polynomial[new_monomial] = 0
 
-                    terms_row_col[new_monomial] += value
+                    polynomial[new_monomial] += value
 
-                if 0 < len(terms_row_col):
-                    terms[poly_row, poly_col] = terms_row_col
+                if 0 < len(polynomial):
+                    poly_matrix_data[poly_row, poly_col] = polynomial
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=left.shape,
         )
 
@@ -105,64 +100,3 @@ class ElemMultExprMixin(ExpressionBaseMixin):
         state, right = self.right.apply(state=state)
 
         return self.elem_mult(state, left, right)
-
-        # if left.shape != right.shape and left.shape == (1, 1):
-        #     left, right = right, left
-
-        # if right.shape == (1, 1):
-        #     right_poly = right.get_poly(0, 0)
-
-        #     @dataclassabc.dataclassabc(frozen=True)
-        #     class BroadCastedPolyMatrix(PolyMatrixMixin):
-        #         underlying: tuple[tuple[int], float]
-        #         shape: tuple[int, int]
-
-        #         def get_poly(self, row: int, col: int) -> typing.Optional[dict[tuple[int, ...], float]]:
-        #             return self.underlying
-
-        #     right = BroadCastedPolyMatrix(
-        #         underlying=right_poly,
-        #         shape=left.shape,
-        #     )
-
-        # terms = {}
-
-        # for poly_row in range(left.shape[0]):
-        #     for poly_col in range(left.shape[1]):
-
-        #         terms_row_col = {}
-
-        #         left_terms = left.get_poly(poly_row, poly_col)
-        #         if left_terms is None:
-        #             continue
-
-        #         right_terms = right.get_poly(poly_row, poly_col)
-        #         if right_terms is None:
-        #             continue
-
-        #         for (left_monomial, left_value), (right_monomial, right_value) \
-        #                 in itertools.product(left_terms.items(), right_terms.items()):
-
-        #             value = left_value * right_value
-
-        #             # if value == 0:
-        #             #     continue
-
-        #             # monomial = tuple(sorted(left_monomial + right_monomial))
-
-        #             new_monomial = merge_monomial_indices((left_monomial, right_monomial))
-
-        #             if new_monomial not in terms_row_col:
-        #                 terms_row_col[new_monomial] = 0
-
-        #             terms_row_col[new_monomial] += value
-
-        #         if 0 < len(terms_row_col):
-        #             terms[poly_row, poly_col] = terms_row_col
-
-        # poly_matrix = init_poly_matrix(
-        #     terms=terms,
-        #     shape=left.shape,
-        # )
-
-        # return state, poly_matrix        
diff --git a/polymatrix/expression/mixins/evalexprmixin.py b/polymatrix/expression/mixins/evalexprmixin.py
index 0c23c1e..b86cafa 100644
--- a/polymatrix/expression/mixins/evalexprmixin.py
+++ b/polymatrix/expression/mixins/evalexprmixin.py
@@ -52,18 +52,18 @@ class EvalExprMixin(ExpressionBaseMixin):
             initial=(state, tuple(), tuple())
         )
         
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
             for col in range(underlying.shape[1]):
             
-                underlying_terms = underlying.get_poly(row, col)
-                if underlying_terms is None:
+                underlying_polynomial = underlying.get_poly(row, col)
+                if underlying_polynomial is None:
                     continue
 
-                terms_row_col = {}
+                polynomial = {}
 
-                for monomial, value in underlying_terms.items():
+                for monomial, value in underlying_polynomial.items():
                     
                     def acc_monomial(acc, next):
                         new_monomial, value = acc
@@ -83,20 +83,20 @@ class EvalExprMixin(ExpressionBaseMixin):
                         initial=(tuple(), value),
                     ))
 
-                    if new_monomial not in terms_row_col:
-                        terms_row_col[new_monomial] = 0
+                    if new_monomial not in polynomial:
+                        polynomial[new_monomial] = 0
 
-                    terms_row_col[new_monomial] += new_value
+                    polynomial[new_monomial] += new_value
 
                     # delete zero entries
-                    if math.isclose(terms_row_col[new_monomial], 0, abs_tol=1e-12):
-                        del terms_row_col[new_monomial]
+                    if math.isclose(polynomial[new_monomial], 0, abs_tol=1e-12):
+                        del polynomial[new_monomial]
 
-                if 0 < len(terms_row_col):
-                    terms[row, col] = terms_row_col
+                if 0 < len(polynomial):
+                    poly_matrix_data[row, col] = polynomial
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/filterexprmixin.py b/polymatrix/expression/mixins/filterexprmixin.py
index 86b5c6a..a240102 100644
--- a/polymatrix/expression/mixins/filterexprmixin.py
+++ b/polymatrix/expression/mixins/filterexprmixin.py
@@ -36,14 +36,14 @@ class FilterExprMixin(ExpressionBaseMixin):
         assert predicator.shape[1] == 1
         assert underlying.shape[0] == predicator.shape[0]
 
-        terms = {}
+        poly_matrix_data = {}
         row_index = 0
 
         for row in range(underlying.shape[0]):
 
-            underlying_terms = underlying.get_poly(row, 0)
+            underlying_polynomial = underlying.get_poly(row, 0)
 
-            if underlying_terms is None:
+            if underlying_polynomial is None:
                 continue
 
             predicator_poly = predicator.get_poly(row, 0)
@@ -64,11 +64,11 @@ class FilterExprMixin(ExpressionBaseMixin):
                     predicator_value = 0
 
             if (predicator_value != 0) is not self.inverse:
-                terms[row_index, 0] = underlying_terms
+                poly_matrix_data[row_index, 0] = underlying_polynomial
                 row_index += 1
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(row_index, 1),
         )
 
diff --git a/polymatrix/expression/mixins/filterlinearpartexprmixin.py b/polymatrix/expression/mixins/filterlinearpartexprmixin.py
index dfca992..562ac61 100644
--- a/polymatrix/expression/mixins/filterlinearpartexprmixin.py
+++ b/polymatrix/expression/mixins/filterlinearpartexprmixin.py
@@ -29,7 +29,7 @@ class FilterLinearPartExprMixin(ExpressionBaseMixin):
         state, variables = self.variables.apply(state=state)
 
         def gen_variable_monomials():
-            for _, term in variables.gen_terms():
+            for _, term in variables.gen_data():
                 assert len(term) == 1, f'{term} should have only a single monomial'
 
                 for monomial in term.keys():
@@ -37,18 +37,18 @@ class FilterLinearPartExprMixin(ExpressionBaseMixin):
 
         variable_monomials = tuple(gen_variable_monomials())
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
             for col in range(underlying.shape[1]):
 
-                underlying_terms = underlying.get_poly(row, col)
-                if underlying_terms is None:
+                underlying_polynomial = underlying.get_poly(row, col)
+                if underlying_polynomial is None:
                     continue
 
-                monomial_terms = collections.defaultdict(float)
+                polynomial = collections.defaultdict(float)
 
-                for monomial, value in underlying_terms.items():
+                for monomial, value in underlying_polynomial.items():
 
                     for variable_monomial in variable_monomials:
 
@@ -63,15 +63,14 @@ class FilterLinearPartExprMixin(ExpressionBaseMixin):
 
                         # take the first that matches
                         if all(variable not in remainder for variable in variable_monomial):
-                            monomial_terms[remainder] += value
+                            polynomial[remainder] += value
 
                             break
 
-                terms[row, col] = monomial_terms
-                # terms[row, col] = dict(monomial_terms)
+                poly_matrix_data[row, col] = dict(polynomial)
         
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/fromsymmetricmatrixexprmixin.py b/polymatrix/expression/mixins/fromsymmetricmatrixexprmixin.py
index 7d48c0f..c7d90a1 100644
--- a/polymatrix/expression/mixins/fromsymmetricmatrixexprmixin.py
+++ b/polymatrix/expression/mixins/fromsymmetricmatrixexprmixin.py
@@ -28,18 +28,18 @@ class FromSymmetricMatrixExprMixin(ExpressionBaseMixin):
 
         assert underlying.shape[0] == underlying.shape[1]
 
-        terms = {}
+        poly_matrix_data = {}
         var_index = 0
 
         for row in range(underlying.shape[0]):
             for col in range(row, underlying.shape[1]):
                 
-                terms[var_index, 0] = underlying.get_poly(row, col)
+                poly_matrix_data[var_index, 0] = underlying.get_poly(row, col)
 
                 var_index += 1
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(var_index, 1),
         )
 
diff --git a/polymatrix/expression/mixins/fromtermsexprmixin.py b/polymatrix/expression/mixins/fromtermsexprmixin.py
index 8aac10b..22512d5 100644
--- a/polymatrix/expression/mixins/fromtermsexprmixin.py
+++ b/polymatrix/expression/mixins/fromtermsexprmixin.py
@@ -26,10 +26,10 @@ class FromTermsExprMixin(ExpressionBaseMixin):
         state: ExpressionStateMixin,
     ) -> tuple[ExpressionStateMixin, PolyMatrixMixin]:
 
-        terms = {coord: dict(monomials) for coord, monomials in self.terms}
+        data = {coord: dict(monomials) for coord, monomials in self.terms}
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=data,
             shape=self.shape,
         )
 
diff --git a/polymatrix/expression/mixins/fromtupleexprmixin.py b/polymatrix/expression/mixins/fromtupleexprmixin.py
index 19b3ed3..92b965c 100644
--- a/polymatrix/expression/mixins/fromtupleexprmixin.py
+++ b/polymatrix/expression/mixins/fromtupleexprmixin.py
@@ -100,7 +100,7 @@ class FromTupleExprMixin(ExpressionBaseMixin):
                 polynomials[poly_row, poly_col] = polynomial
 
         poly_matrix = init_poly_matrix(
-            terms=polynomials,
+            data=polynomials,
             shape=(len(self.data), len(self.data[0])),
         )
 
diff --git a/polymatrix/expression/mixins/halfnewtonpolytopeexprmixin.py b/polymatrix/expression/mixins/halfnewtonpolytopeexprmixin.py
index 21d16fd..445e776 100644
--- a/polymatrix/expression/mixins/halfnewtonpolytopeexprmixin.py
+++ b/polymatrix/expression/mixins/halfnewtonpolytopeexprmixin.py
@@ -106,7 +106,7 @@ class HalfNewtonPolytopeExprMixin(ExpressionBaseMixin):
         )
 
         poly_matrix = init_poly_matrix(
-            terms={(row, 0): {monom: 1} for row, monom in enumerate(monomials)},
+            data={(row, 0): {monom: 1} for row, monom in enumerate(monomials)},
             shape=(len(monomials), 1),
         )
 
diff --git a/polymatrix/expression/mixins/legendreseriesmixin.py b/polymatrix/expression/mixins/legendreseriesmixin.py
index 194e65e..aa3de58 100644
--- a/polymatrix/expression/mixins/legendreseriesmixin.py
+++ b/polymatrix/expression/mixins/legendreseriesmixin.py
@@ -36,27 +36,27 @@ class LegendreSeriesMixin(ExpressionBaseMixin):
         else:
             degrees = self.degrees
         
-        terms = {}
+        poly_matrix_data = {}
 
         for degree in degrees:
         # for degree in self.degree:
             poly = underlying.get_poly(degree, 0)
 
-            terms[degree, 0] = dict(poly)
+            poly_matrix_data[degree, 0] = dict(poly)
 
             if 2 <= degree:
                 poly = underlying.get_poly(degree - 2, 0)
                 factor = - (degree - 1) / (degree + 1)
 
                 for m, v in poly.items():
-                    if m in terms[degree, 0]:
-                        terms[degree, 0][m] += v*factor
+                    if m in poly_matrix_data[degree, 0]:
+                        poly_matrix_data[degree, 0][m] += v*factor
                     else:
-                        terms[degree, 0][m] = v*factor
+                        poly_matrix_data[degree, 0][m] = v*factor
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
-            shape=(len(terms), 1),
+            data=poly_matrix_data,
+            shape=(len(poly_matrix_data), 1),
         )
 
         return state, poly_matrix
diff --git a/polymatrix/expression/mixins/linearinexprmixin.py b/polymatrix/expression/mixins/linearinexprmixin.py
index ded61ec..f0b57d3 100644
--- a/polymatrix/expression/mixins/linearinexprmixin.py
+++ b/polymatrix/expression/mixins/linearinexprmixin.py
@@ -87,7 +87,7 @@ class LinearInExprMixin(ExpressionBaseMixin):
                 terms[row, col][p_monomial] = value
         
         poly_matrix = init_poly_matrix(
-            terms=dict(terms),
+            data=dict(terms),
             shape=(underlying.shape[0], len(monomials)),
         )
 
diff --git a/polymatrix/expression/mixins/linearmatrixinexprmixin.py b/polymatrix/expression/mixins/linearmatrixinexprmixin.py
index 4d25a1e..3bb8dfe 100644
--- a/polymatrix/expression/mixins/linearmatrixinexprmixin.py
+++ b/polymatrix/expression/mixins/linearmatrixinexprmixin.py
@@ -53,7 +53,7 @@ class LinearMatrixInExprMixin(ExpressionBaseMixin):
                             break
         
         poly_matrix = init_poly_matrix(
-            terms=dict(terms),
+            data=dict(terms),
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/linearmonomialsexprmixin.py b/polymatrix/expression/mixins/linearmonomialsexprmixin.py
index ed9e79f..c7de7f1 100644
--- a/polymatrix/expression/mixins/linearmonomialsexprmixin.py
+++ b/polymatrix/expression/mixins/linearmonomialsexprmixin.py
@@ -62,14 +62,14 @@ class LinearMonomialsExprMixin(ExpressionBaseMixin):
 
         linear_monomials = sort_monomials(set(gen_linear_monomials()))
 
-        def gen_terms():
+        def gen_data():
             for index, monomial in enumerate(linear_monomials):
                 yield (index, 0), {monomial: 1.0}
 
-        terms = dict(gen_terms())
+        poly_matrix_data = dict(gen_data())
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(len(linear_monomials), 1),
         )
 
diff --git a/polymatrix/expression/mixins/matrixmultexprmixin.py b/polymatrix/expression/mixins/matrixmultexprmixin.py
index b7ae5ce..22a6c94 100644
--- a/polymatrix/expression/mixins/matrixmultexprmixin.py
+++ b/polymatrix/expression/mixins/matrixmultexprmixin.py
@@ -42,30 +42,30 @@ class MatrixMultExprMixin(ExpressionBaseMixin):
                 stack=self.stack,
             ))
 
-        terms = {}
+        poly_matrix_data = {}
 
         for poly_row in range(left.shape[0]):
             for poly_col in range(right.shape[1]):
 
-                terms_row_col = {}
+                polynomial = {}
 
                 for index_k in range(left.shape[1]):
 
-                    left_terms = left.get_poly(poly_row, index_k)
-                    if left_terms is None:
+                    left_polynomial = left.get_poly(poly_row, index_k)
+                    if left_polynomial is None:
                         continue
 
-                    right_terms = right.get_poly(index_k, poly_col)
-                    if right_terms is None:
+                    right_polynomial = right.get_poly(index_k, poly_col)
+                    if right_polynomial is None:
                         continue
 
-                    multiply_polynomial(left_terms, right_terms, terms_row_col)
+                    multiply_polynomial(left_polynomial, right_polynomial, polynomial)
 
-                if 0 < len(terms_row_col):
-                    terms[poly_row, poly_col] = terms_row_col
+                if 0 < len(polynomial):
+                    poly_matrix_data[poly_row, poly_col] = polynomial
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(left.shape[0], right.shape[1]),
         )
 
diff --git a/polymatrix/expression/mixins/maxexprmixin.py b/polymatrix/expression/mixins/maxexprmixin.py
index bfabc4d..fa1155e 100644
--- a/polymatrix/expression/mixins/maxexprmixin.py
+++ b/polymatrix/expression/mixins/maxexprmixin.py
@@ -21,7 +21,7 @@ class MaxExprMixin(ExpressionBaseMixin):
 
         state, underlying = self.underlying.apply(state)
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
 
@@ -37,10 +37,10 @@ class MaxExprMixin(ExpressionBaseMixin):
             values = tuple(gen_values())
 
             if 0 < len(values):
-                terms[row, 0] = {tuple(): max(values)}
+                poly_matrix_data[row, 0] = {tuple(): max(values)}
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(underlying.shape[0], 1),
         )
 
diff --git a/polymatrix/expression/mixins/parametrizeexprmixin.py b/polymatrix/expression/mixins/parametrizeexprmixin.py
index 712ff3f..0160b9a 100644
--- a/polymatrix/expression/mixins/parametrizeexprmixin.py
+++ b/polymatrix/expression/mixins/parametrizeexprmixin.py
@@ -50,14 +50,14 @@ class ParametrizeExprMixin(ExpressionBaseMixin):
 
         # assert underlying.shape[1] == 1
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
             var_index = start + row
-            terms[row, 0] = {((var_index, 1),): 1}
+            poly_matrix_data[row, 0] = {((var_index, 1),): 1}
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/parametrizematrixexprmixin.py b/polymatrix/expression/mixins/parametrizematrixexprmixin.py
index 1aa393b..c51bcf0 100644
--- a/polymatrix/expression/mixins/parametrizematrixexprmixin.py
+++ b/polymatrix/expression/mixins/parametrizematrixexprmixin.py
@@ -33,13 +33,13 @@ class ParametrizeMatrixExprMixin(ExpressionBaseMixin):
 
         assert underlying.shape[1] == 1
 
-        terms = {}
+        poly_matrix_data = {}
         var_index = 0
 
         for row in range(underlying.shape[0]):
             for _ in range(row, underlying.shape[0]):
                 
-                terms[var_index, 0] = {((state.n_param + var_index, 1),): 1.0}
+                poly_matrix_data[var_index, 0] = {((state.n_param + var_index, 1),): 1.0}
 
                 var_index += 1
 
@@ -49,7 +49,7 @@ class ParametrizeMatrixExprMixin(ExpressionBaseMixin):
         )
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(var_index, 1),
         )
 
diff --git a/polymatrix/expression/mixins/productexprmixin.py b/polymatrix/expression/mixins/productexprmixin.py
index 0168e27..b41865b 100644
--- a/polymatrix/expression/mixins/productexprmixin.py
+++ b/polymatrix/expression/mixins/productexprmixin.py
@@ -1,6 +1,7 @@
 
 import abc
 import itertools
+from polymatrix.polymatrix.typing import PolynomialData
 from polymatrix.polymatrix.utils.multiplypolynomial import multiply_polynomial
 
 from polymatrix.utils.getstacklines import FrameSummary
@@ -31,23 +32,9 @@ class ProductExprMixin(ExpressionBaseMixin):
         self, 
         state: ExpressionState,
     ) -> tuple[ExpressionState, PolyMatrix]:
-        
-        # if self.number == 0:
-        #     terms = {(0, 0): {tuple(): 1.0}}
-
-        #     poly_matrix = init_poly_matrix(
-        #         terms=terms,
-        #         shape=(1, 1),
-        #     )
-
-        # elif self.number == 1:
-        #     state, monomials = self.monomials.apply(state=state) 
-        #     poly_matrix = monomials
-
-        # else:
 
         if len(self.underlying) == 0:
-            terms = {(0,0): {tuple(): 1}}
+            poly_matrix_data = {(0,0): {tuple(): 1}}
         
         else:
 
@@ -65,23 +52,6 @@ class ProductExprMixin(ExpressionBaseMixin):
                 initial=(state, tuple())
             )
 
-            # highest_degrees = tuple(e.shape[0] for e in underlying)
-
-            # if self.degrees is None:
-            #     degrees = range(sum(highest_degrees))
-
-            # else:
-            #     degrees = self.degrees
-
-            # max_degree = max(degrees)
-
-            # for poly_matrix in underlying:
-            #     if not (max_degree <= poly_matrix.shape[0]):
-            #         raise AssertionError(to_operator_exception(
-            #             message=f'{poly_matrix.shape[0]} < {max_degree}',
-            #             stack=self.stack,
-            #         ))
-
             def gen_indices():
                 product_indices = itertools.product(*(range(e.shape[0]) for e in underlying))
 
@@ -92,26 +62,16 @@ class ProductExprMixin(ExpressionBaseMixin):
                     yield from filter(lambda v: sum(v) in self.degrees, product_indices)
                     
             indices = tuple(gen_indices())
-            # print(indices)
-
-            # indices = filter(lambda v: sum(v) <= self.degree, itertools.product(*(range(self.degree) for _ in range(dim))))
 
-            terms = {}
+            poly_matrix_data = {}
 
             for row, indexing in enumerate(indices):
 
-                # def acc_product(acc, v):
-                #     left_monomials = acc
-                #     polymatrix, row = v
-
-                #     right_monomials = polymatrix.get_poly(row, 0).keys()
-
-                #     if left_monomials is (None,):
-                #         return right_monomials
-
-                #     return tuple(multiply_monomials(left_monomials, right_monomials))
-
-                def acc_product(left, v):
+                def acc_product(
+                        left: PolynomialData, 
+                        v: tuple[ExpressionBaseMixin, int],
+                ) -> PolynomialData:
+                    
                     poly_matrix, row = v
 
                     right = poly_matrix.get_poly(row, 0)
@@ -129,11 +89,11 @@ class ProductExprMixin(ExpressionBaseMixin):
                     initial={},
                 )
 
-                terms[row, 0] = polynomial
+                poly_matrix_data[row, 0] = polynomial
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
-            shape=(len(terms), 1),
+            data=poly_matrix_data,
+            shape=(len(poly_matrix_data), 1),
         )
 
         return state, poly_matrix   
diff --git a/polymatrix/expression/mixins/quadraticinexprmixin.py b/polymatrix/expression/mixins/quadraticinexprmixin.py
index e739364..2cc9e12 100644
--- a/polymatrix/expression/mixins/quadraticinexprmixin.py
+++ b/polymatrix/expression/mixins/quadraticinexprmixin.py
@@ -87,7 +87,7 @@ class QuadraticInExprMixin(ExpressionBaseMixin):
             terms[row, col][p_monomial] += value
 
         poly_matrix = init_poly_matrix(
-            terms=dict((k, dict(v)) for k, v in terms.items()),
+            data=dict((k, dict(v)) for k, v in terms.items()),
             shape=2*(len(sos_monomials),),
         )
 
diff --git a/polymatrix/expression/mixins/quadraticmonomialsexprmixin.py b/polymatrix/expression/mixins/quadraticmonomialsexprmixin.py
index 99befc8..3a7de30 100644
--- a/polymatrix/expression/mixins/quadraticmonomialsexprmixin.py
+++ b/polymatrix/expression/mixins/quadraticmonomialsexprmixin.py
@@ -65,14 +65,14 @@ class QuadraticMonomialsExprMixin(ExpressionBaseMixin):
 
         sos_monomials = tuple(sorted(set(gen_sos_monomials()), key=lambda m: (len(m), m)))
 
-        def gen_terms():
+        def gen_data():
             for index, monomial in enumerate(sos_monomials):
                 yield (index, 0), {monomial: 1.0}
 
-        terms = dict(gen_terms())
+        poly_matrix_data = dict(gen_data())
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(len(sos_monomials), 1),
         )
 
diff --git a/polymatrix/expression/mixins/squeezeexprmixin.py b/polymatrix/expression/mixins/squeezeexprmixin.py
index eed3d71..888300f 100644
--- a/polymatrix/expression/mixins/squeezeexprmixin.py
+++ b/polymatrix/expression/mixins/squeezeexprmixin.py
@@ -23,7 +23,7 @@ class SqueezeExprMixin(ExpressionBaseMixin):
 
         assert underlying.shape[1] == 1
 
-        terms = {}
+        poly_matrix_data = {}
         row_index = 0
 
         for row in range(underlying.shape[0]):
@@ -32,18 +32,18 @@ class SqueezeExprMixin(ExpressionBaseMixin):
             if polynomial is None:
                 continue
 
-            terms_row_col = {}
+            polynomial = {}
 
             for monomial, value in polynomial.items():
                 if value != 0.0:
-                    terms_row_col[monomial] = value
+                    polynomial[monomial] = value
 
-            if len(terms_row_col):
-                terms[row_index, 0] = terms_row_col
+            if len(polynomial):
+                poly_matrix_data[row_index, 0] = polynomial
                 row_index += 1
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(row_index, 1),
         )
 
diff --git a/polymatrix/expression/mixins/substituteexprmixin.py b/polymatrix/expression/mixins/substituteexprmixin.py
index b75a2de..2bc36d1 100644
--- a/polymatrix/expression/mixins/substituteexprmixin.py
+++ b/polymatrix/expression/mixins/substituteexprmixin.py
@@ -67,7 +67,7 @@ class SubstituteExprMixin(ExpressionBaseMixin):
         else:
             assert len(variable_indices) == len(substitutions), f'{substitutions=}'
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
             for col in range(underlying.shape[1]):
@@ -76,11 +76,11 @@ class SubstituteExprMixin(ExpressionBaseMixin):
                 if polynomial is None:
                     continue
 
-                terms_row_col = collections.defaultdict(float)
+                polynomial = collections.defaultdict(float)
 
                 for monomial, value in polynomial.items():
 
-                    terms_monomial = {tuple(): value}
+                    substituted_monomial = {tuple(): value}
 
                     for variable, count in monomial:
                         if variable in variable_indices:
@@ -90,24 +90,24 @@ class SubstituteExprMixin(ExpressionBaseMixin):
                             for _ in range(count):
                         
                                 next = {}
-                                multiply_polynomial(terms_monomial, substitution, next)
-                                terms_monomial = next
+                                multiply_polynomial(substituted_monomial, substitution, next)
+                                substituted_monomial = next
 
                         else:
                             next = {}
-                            multiply_polynomial(terms_monomial, {((variable, count),): 1.0}, next)
-                            terms_monomial = next
+                            multiply_polynomial(substituted_monomial, {((variable, count),): 1.0}, next)
+                            substituted_monomial = next
 
-                    for monomial, value in terms_monomial.items():
-                        terms_row_col[monomial] += value
+                    for monomial, value in substituted_monomial.items():
+                        polynomial[monomial] += value
 
-                terms_row_col = {key: val for key, val in terms_row_col.items() if not math.isclose(val, 0, abs_tol=1e-12)}
+                polynomial = {key: val for key, val in polynomial.items() if not math.isclose(val, 0, abs_tol=1e-12)}
 
-                if 0 < len(terms_row_col):
-                    terms[row, col] = terms_row_col
+                if 0 < len(polynomial):
+                    poly_matrix_data[row, col] = polynomial
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/subtractmonomialsexprmixin.py b/polymatrix/expression/mixins/subtractmonomialsexprmixin.py
index 9275332..a01b5e7 100644
--- a/polymatrix/expression/mixins/subtractmonomialsexprmixin.py
+++ b/polymatrix/expression/mixins/subtractmonomialsexprmixin.py
@@ -44,10 +44,10 @@ class SubtractMonomialsExprMixin(ExpressionBaseMixin):
 
         remainders = sort_monomials(set(gen_remainders()))
 
-        terms = {(row, 0): {remainder: 1.0} for row, remainder in enumerate(remainders)}            
+        data = {(row, 0): {remainder: 1.0} for row, remainder in enumerate(remainders)}            
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=data,
             shape=(len(remainders), 1),
         )
 
diff --git a/polymatrix/expression/mixins/sumexprmixin.py b/polymatrix/expression/mixins/sumexprmixin.py
index 812b260..83f9397 100644
--- a/polymatrix/expression/mixins/sumexprmixin.py
+++ b/polymatrix/expression/mixins/sumexprmixin.py
@@ -29,7 +29,7 @@ class SumExprMixin(ExpressionBaseMixin):
 
         state, underlying = self.underlying.apply(state)
 
-        terms = collections.defaultdict(dict)
+        poly_matrix_data = collections.defaultdict(dict)
 
         for row in range(underlying.shape[0]):
             for col in range(underlying.shape[1]):
@@ -38,16 +38,16 @@ class SumExprMixin(ExpressionBaseMixin):
                 if polynomial is None:
                     continue
 
-                term_monomials = terms[row, 0]
+                polynomial = poly_matrix_data[row, 0]
 
                 for monomial, value in polynomial.items():
-                    if monomial in term_monomials:
-                        term_monomials[monomial] += value
+                    if monomial in polynomial:
+                        polynomial[monomial] += value
                     else:
-                        term_monomials[monomial] = value
+                        polynomial[monomial] = value
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(underlying.shape[0], 1),
         )
 
diff --git a/polymatrix/expression/mixins/toconstantexprmixin.py b/polymatrix/expression/mixins/toconstantexprmixin.py
index b8b3d64..c3dc4e2 100644
--- a/polymatrix/expression/mixins/toconstantexprmixin.py
+++ b/polymatrix/expression/mixins/toconstantexprmixin.py
@@ -39,7 +39,7 @@ class ToConstantExprMixin(ExpressionBaseMixin):
                     terms[row, col][tuple()] = polynomial[tuple()]
         
         poly_matrix = init_poly_matrix(
-            terms=dict(terms),
+            data=dict(terms),
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/toquadraticexprmixin.py b/polymatrix/expression/mixins/toquadraticexprmixin.py
index d10fcd9..c6b692b 100644
--- a/polymatrix/expression/mixins/toquadraticexprmixin.py
+++ b/polymatrix/expression/mixins/toquadraticexprmixin.py
@@ -9,7 +9,6 @@ from polymatrix.polymatrix.abc import PolyMatrix
 from polymatrix.expressionstate.abc import ExpressionState
 
 
-# to be deleted?
 class ToQuadraticExprMixin(ExpressionBaseMixin):
     @property
     @abc.abstractmethod
@@ -25,17 +24,17 @@ class ToQuadraticExprMixin(ExpressionBaseMixin):
 
         state = [state]
 
-        terms = {}
+        poly_matrix_data = {}
         auxillary_equations_from_quadratic = {}
 
         def to_quadratic(monomial_terms):
-            terms_row_col = collections.defaultdict(float)
+            polynomial = collections.defaultdict(float)
 
             for monomial, value in monomial_terms.items():
                 
                 if 2 < len(monomial):
                     current_aux = state[0].n_param
-                    terms_row_col[(monomial[0], current_aux)] += value
+                    polynomial[(monomial[0], current_aux)] += value
                     state[0] = state[0].register(n_param=1)
 
                     for variable in monomial[1:-2]:
@@ -52,10 +51,10 @@ class ToQuadraticExprMixin(ExpressionBaseMixin):
                     }
 
                 else:                       
-                    terms_row_col[monomial] += value
+                    polynomial[monomial] += value
 
             # return dict(terms_row_col)
-            return terms_row_col
+            return polynomial
 
         for row in range(underlying.shape[0]):
             for col in range(underlying.shape[1]):
@@ -64,43 +63,18 @@ class ToQuadraticExprMixin(ExpressionBaseMixin):
                 if polynomial is None:
                     continue
 
-                terms_row_col = to_quadratic(
+                polynomial = to_quadratic(
                     monomial_terms=polynomial,
                 )
 
-                # terms_row_col = collections.defaultdict(float)
-
-                # for monomial, value in underlying_terms.items():
-                    
-                #     if 2 < len(monomial):
-                #         current_aux = state.n_param
-                #         terms_row_col[(monomial[0], current_aux)] += value
-                #         state = state.register(n_param=1)
-
-                #         for variable in monomial[1:-2]:
-                #             auxillary_equations[current_aux] = {
-                #                 (variable, current_aux + 1): 1,
-                #                 (current_aux,): -1,
-                #             }
-                #             state = state.register(n_param=1)
-                #             current_aux += 1
-
-                #         auxillary_equations[current_aux] = {
-                #             (monomial[-2], monomial[-1]): 1,
-                #             (current_aux,): -1,
-                #         }
-
-                #     else:                       
-                #         terms_row_col[monomial] += value
-
-                terms[row, col] = terms_row_col
+                poly_matrix_data[row, col] = polynomial
 
         def gen_auxillary_equations():
             for key, monomial_terms in state[0].auxillary_equations.items():
-                terms_row_col = to_quadratic(
+                polynomial = to_quadratic(
                     monomial_terms=monomial_terms,
                 )
-                yield key, terms_row_col
+                yield key, polynomial
 
         state = dataclasses.replace(
             state[0],
@@ -108,7 +82,7 @@ class ToQuadraticExprMixin(ExpressionBaseMixin):
         )
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/mixins/tosortedvariablesmixin.py b/polymatrix/expression/mixins/tosortedvariablesmixin.py
index a61a7e9..32e2162 100644
--- a/polymatrix/expression/mixins/tosortedvariablesmixin.py
+++ b/polymatrix/expression/mixins/tosortedvariablesmixin.py
@@ -31,7 +31,7 @@ class ToSortedVariablesExprMixin(ExpressionBaseMixin):
                 yield (row, 0), {((index, 1),): 1}
         
         poly_matrix = init_poly_matrix(
-            terms=dict(gen_sorted_vector()),
+            data=dict(gen_sorted_vector()),
             shape=(len(variable_indices), 1),
         )
 
diff --git a/polymatrix/expression/mixins/tosymmetricmatrixexprmixin.py b/polymatrix/expression/mixins/tosymmetricmatrixexprmixin.py
index 8801147..488c991 100644
--- a/polymatrix/expression/mixins/tosymmetricmatrixexprmixin.py
+++ b/polymatrix/expression/mixins/tosymmetricmatrixexprmixin.py
@@ -44,20 +44,20 @@ class ToSymmetricMatrixExprMixin(ExpressionBaseMixin):
 
         n_row = invert_binomial_coefficient(underlying.shape[0])
 
-        terms = {}
+        poly_matrix_data = {}
         var_index = 0
 
         for row in range(n_row):
             for col in range(row, n_row):
-                terms[row, col] = underlying.get_poly(var_index, 0)
+                poly_matrix_data[row, col] = underlying.get_poly(var_index, 0)
 
                 if row != col:
-                    terms[col, row] = terms[row, col]
+                    poly_matrix_data[col, row] = poly_matrix_data[row, col]
 
                 var_index += 1
 
         poly_matrix = init_poly_matrix(
-            terms=terms,
+            data=poly_matrix_data,
             shape=(n_row, n_row),
         )
 
diff --git a/polymatrix/expression/mixins/truncateexprmixin.py b/polymatrix/expression/mixins/truncateexprmixin.py
index fdf3921..2fcb970 100644
--- a/polymatrix/expression/mixins/truncateexprmixin.py
+++ b/polymatrix/expression/mixins/truncateexprmixin.py
@@ -43,7 +43,7 @@ class TruncateExprMixin(ExpressionBaseMixin):
             state, variable_indices = get_variable_indices_from_variable(state, self.variables)
             cond = lambda idx: idx in variable_indices
 
-        terms = {}
+        poly_matrix_data = {}
 
         for row in range(underlying.shape[0]):
             for col in range(underlying.shape[1]):
@@ -52,19 +52,19 @@ class TruncateExprMixin(ExpressionBaseMixin):
                 if polynomial is None:
                     continue
 
-                terms_row_col = {}
+                polynomial = {}
 
                 for monomial, value in polynomial.items():
 
                     degree = sum((count for var_idx, count in monomial if cond(var_idx)))
 
                     if (degree in self.degrees) is not self.inverse:
-                        terms_row_col[monomial] = value
+                        polynomial[monomial] = value
                 
-                terms[row, col] = terms_row_col
+                poly_matrix_data[row, col] = polynomial
         
         poly_matrix = init_poly_matrix(
-            terms=dict(terms),
+            data=dict(poly_matrix_data),
             shape=underlying.shape,
         )
 
diff --git a/polymatrix/expression/to.py b/polymatrix/expression/to.py
index b02bb11..cebc75a 100644
--- a/polymatrix/expression/to.py
+++ b/polymatrix/expression/to.py
@@ -31,7 +31,7 @@ def to_constant(
 
         A = np.zeros(underlying.shape, dtype=np.double)
 
-        for (row, col), polynomial in underlying.gen_terms():
+        for (row, col), polynomial in underlying.gen_data():
             for monomial, value in polynomial.items():
                 if len(monomial) == 0:
                     A[row, col] = value
@@ -86,7 +86,7 @@ def to_sympy(
 
         A = np.zeros(underlying.shape, dtype=object)
 
-        for (row, col), polynomial in underlying.gen_terms():
+        for (row, col), polynomial in underlying.gen_data():
 
             sympy_polynomial = 0
 
diff --git a/polymatrix/expression/utils/broadcastpolymatrix.py b/polymatrix/expression/utils/broadcastpolymatrix.py
new file mode 100644
index 0000000..e8b3ff3
--- /dev/null
+++ b/polymatrix/expression/utils/broadcastpolymatrix.py
@@ -0,0 +1,34 @@
+from polymatrix.polymatrix.init import init_broadcast_poly_matrix, init_poly_matrix
+from polymatrix.utils.getstacklines import FrameSummary
+from polymatrix.utils.tooperatorexception import to_operator_exception
+from polymatrix.polymatrix.abc import PolyMatrix
+
+
+def broadcast_poly_matrix(
+        left: PolyMatrix, 
+        right: PolyMatrix, 
+        stack: tuple[FrameSummary],
+) -> PolyMatrix:
+    
+    # broadcast left
+    if left.shape == (1, 1) and right.shape != (1, 1):
+        left = init_broadcast_poly_matrix(
+            data=left.get_poly(0, 0),
+            shape=right.shape,
+        )
+
+    # broadcast right
+    elif left.shape != (1, 1) and right.shape == (1, 1):
+        right = init_broadcast_poly_matrix(
+            data=right.get_poly(0, 0),
+            shape=left.shape,
+        )
+
+    else:
+        if not (left.shape == right.shape):
+            raise AssertionError(to_operator_exception(
+                message=f'{left.shape} != {right.shape}',
+                stack=stack,
+            ))
+
+    return left, right
diff --git a/polymatrix/polymatrix/impl.py b/polymatrix/polymatrix/impl.py
index f957685..04a7872 100644
--- a/polymatrix/polymatrix/impl.py
+++ b/polymatrix/polymatrix/impl.py
@@ -7,11 +7,11 @@ from polymatrix.polymatrix.typing import PolynomialData
 
 @dataclassabc.dataclassabc(frozen=True)
 class PolyMatrixImpl(PolyMatrix):
-	terms: dict[tuple[int, int], PolynomialData]
+	data: dict[tuple[int, int], PolynomialData]
 	shape: tuple[int, int]
 
 
 @dataclassabc.dataclassabc(frozen=True)
 class BroadcastPolyMatrixImpl(BroadcastPolyMatrixMixin):
-	polynomial: tuple[tuple[int], float]
+	data: tuple[tuple[int], float]
 	shape: tuple[int, int]
diff --git a/polymatrix/polymatrix/init.py b/polymatrix/polymatrix/init.py
index 2dc2a63..fa7bd20 100644
--- a/polymatrix/polymatrix/init.py
+++ b/polymatrix/polymatrix/init.py
@@ -1,13 +1,24 @@
-from polymatrix.polymatrix.impl import PolyMatrixImpl
+from polymatrix.polymatrix.impl import BroadcastPolyMatrixImpl, PolyMatrixImpl
 from polymatrix.polymatrix.typing import PolynomialData
 
 
 def init_poly_matrix(
-		terms: dict[tuple[int, int], PolynomialData],
+		data: dict[tuple[int, int], PolynomialData],
 		shape: tuple[int, int],
 ):
 
 	return PolyMatrixImpl(
-		terms=terms,
+		data=data,
+		shape=shape,
+)
+
+
+def init_broadcast_poly_matrix(
+		data: PolynomialData,
+		shape: tuple[int, int],
+):
+
+	return BroadcastPolyMatrixImpl(
+		data=data,
 		shape=shape,
 )
diff --git a/polymatrix/polymatrix/mixins.py b/polymatrix/polymatrix/mixins.py
index 5046492..0ce4dad 100644
--- a/polymatrix/polymatrix/mixins.py
+++ b/polymatrix/polymatrix/mixins.py
@@ -10,7 +10,7 @@ class PolyMatrixMixin(abc.ABC):
     def shape(self) -> tuple[int, int]:
         ...
 
-    def gen_terms(self) -> typing.Generator[tuple[tuple[int, int], PolynomialData], None, None]:
+    def gen_data(self) -> typing.Generator[tuple[tuple[int, int], PolynomialData], None, None]:
         for row in range(self.shape[0]):
             for col in range(self.shape[1]):
                 polynomial = self.get_poly(row, col)
@@ -30,13 +30,13 @@ class PolyMatrixAsDictMixin(
 ):
     @property
     @abc.abstractmethod
-    def terms(self) -> dict[tuple[int, int], PolynomialData]:
+    def data(self) -> dict[tuple[int, int], PolynomialData]:
         ...
 
     # overwrites the abstract method of `PolyMatrixMixin`
     def get_poly(self, row: int, col: int) -> PolynomialData | None:
-        if (row, col) in self.terms:
-            return self.terms[row, col]
+        if (row, col) in self.data:
+            return self.data[row, col]
 
 
 class BroadcastPolyMatrixMixin(
@@ -45,9 +45,9 @@ class BroadcastPolyMatrixMixin(
 ):
     @property
     @abc.abstractmethod
-    def polynomial(self) -> PolynomialData:
+    def data(self) -> PolynomialData:
         ...
 
     # overwrites the abstract method of `PolyMatrixMixin`
     def get_poly(self, col: int, row: int) -> PolynomialData | None:
-        return self.polynomial
+        return self.data
-- 
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