Documentation

groebner::spoly

The S-polynomial of two polynomials

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Syntax

groebner::spoly(p1, p2, <order>)

Description

groebner::spoly(p1, p2) computes the S-polynomial of the polynomials p1 and p2.

The rules laid down in the introduction to groebner concerning the polynomial types and the ordering apply.

The polynomials must be of the same type. In particular, do not mix polynomials created via poly and polynomial expressions!

Examples

Example 1

The polynomials

p1 := poly(x^2 - x + 2*y^2, [x, y]):
p2 := poly(x + 2*y - 1, [x, y]):

generate the following S-polynomial with respect to lexicographical ordering:

groebner::spoly(p1, p2, LexOrder) delete p1, p2:

Parameters

 p1, p2 A list or set of polynomials or polynomial expressions of the same type. The coefficients in these polynomials and polynomial expressions can be arbitrary arithmetical expressions. order One of the identifiers DegInvLexOrder, DegreeOrder, and LexOrder, or a user-defined term ordering of type Dom::MonomOrdering. The default ordering is DegInvLexOrder.

Return Values

Polynomial of the same type as the input polynomials. If polynomial expressions are used as input, then a polynomial expression is returned.

Algorithms

The S-polynomial of two polynomials p1, p2 is defined to be ,

where lterm and lmonomial are used in the same sense as the MuPAD® functions of the same name. This formula is constructed such that the leading terms of the two summands cancel.