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Little addition to polynomials design note.

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Gleb Minaev 2022-06-26 12:58:30 +03:00
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@ -8,7 +8,7 @@ There are 3 approaches to represent polynomials:
1. For univariate polynomials one can represent and store polynomial as a list of coefficients for each power of the variable. I.e. polynomial $a_0 + \dots + a_n x^n $ can be represented as a finite sequence $(a_0; \dots; a_n)$. (Compare to sequential definition of polynomials.) 1. For univariate polynomials one can represent and store polynomial as a list of coefficients for each power of the variable. I.e. polynomial $a_0 + \dots + a_n x^n $ can be represented as a finite sequence $(a_0; \dots; a_n)$. (Compare to sequential definition of polynomials.)
2. For multivariate polynomials one can represent and store polynomial as a matching (in programming it is called "map" or "dictionary", in math it is called [functional relation](https://en.wikipedia.org/wiki/Binary_relation#Special_types_of_binary_relations)) of each "**term signature**" (that describes what variables and in what powers appear in the term) with corresponding coefficient of the term. But there are 2 possible approaches of term signature representation: 2. For multivariate polynomials one can represent and store polynomial as a matching (in programming it is called "map" or "dictionary", in math it is called [functional relation](https://en.wikipedia.org/wiki/Binary_relation#Special_types_of_binary_relations)) of each "**term signature**" (that describes what variables and in what powers appear in the term) with corresponding coefficient of the term. But there are 2 possible approaches of term signature representation:
1. One can number all the variables, so term signature can be represented as a sequence describing powers of the variables. I.e. signature of term $c \\; x_0^{d_0} \dots x_n^{d_n} $ (for natural or zero $d_i $) can be represented as a finite sequence $(d_0; \dots; d_n)$. 1. One can number all the variables, so term signature can be represented as a sequence describing powers of the variables. I.e. signature of term $c \\; x_0^{d_0} \dots x_n^{d_n} $ (for natural or zero $d_i $) can be represented as a finite sequence $(d_0; \dots; d_n)$.
2. One can represent variables as objects ("**labels**"), so term signature can be also represented as a matching of each appeared variable with its power in the term. 2. One can represent variables as objects ("**labels**"), so term signature can be also represented as a matching of each appeared variable with its power in the term. I.e. signature of term $c \\; x_0^{d_0} \dots x_n^{d_n} $ (for natural non-zero $d_i $) can be represented as a finite matching $(x_0 \to d_1; \dots; x_n \to d_n)$.
All that three approaches are implemented by "list", "numbered", and "labeled" versions of polynomials and polynomial spaces respectively. Whereas all rational functions are represented as fractions with corresponding polynomial numerator and denominator, and rational functions' spaces are implemented in the same way as usual field of rational numbers (or more precisely, as any field of fractions over integral domain) should be implemented. All that three approaches are implemented by "list", "numbered", and "labeled" versions of polynomials and polynomial spaces respectively. Whereas all rational functions are represented as fractions with corresponding polynomial numerator and denominator, and rational functions' spaces are implemented in the same way as usual field of rational numbers (or more precisely, as any field of fractions over integral domain) should be implemented.