Glossary¶

polynomial

http://eom.springer.de/P/p073690.htm

admissible ordering of monomials

An ordering, >, on \N^n is called an admissible ordering of monomials when it satisfies the following conditions:

1. > is a total ordering on \N^n
2. \alpha, \beta, \gamma \in \N^n and \alpha > \beta, then \alpha + \gamma > \beta + \gamma
3. \alpha \ge 0 for all \alpha \in \N^n

algebraic variety

The set of solutions to a system of polynomial equations.

characteristic zero

A property of algebraic structures. An algebraic structure is of characteristic zero if it has infinite number of elements. Trivial examples are the ring of integers or the field of rational numbers. Contrary, an algebraic structure can be of positive characteristic, if it has finite number of elements. Then the characteristic tells how many elements are in the structure. In SymPy we can check if a domain has zero characteristic using has_CharacteristicZero.

symmetric polynomial

A polynomial f \in \R\Xn, where \R is a ring, is called symmetric if \sigma(f) = f holds for every permutation \sigma of the set \{x_1, \ldots, x_n\}. Every symmetric polynomial can be rewritten in terms of elementary symmetric polynomials utilizing method called symmetric reduction. This can can be accomplished in SymPy using symmetrize() function.

elementary symmetric polynomial

A polynomial f \in \R\Xn, where \R is a ring, is called an elementary symmetric polynomial if f is symmetric and f is an element of the basis which generates all symmetric polynomials. The k–th elementary symmetric polynomial in n variables, i.e. a polynomial of degree k, can be constructed by summing all different k–th degree monomials in \{x_1, \ldots, x_n\}. Elementary symmetric polynomials can be generated in SymPy using symmetric_poly() function.

monic polynomial

A polynomial with the leading coefficient equal to the identity element of the ground domain, or to zero if the polynomial itself is zero. In SymPy this can can be checked using is_monic property on an instance of Poly class.