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groebner::normalf -- complete reduction modulo a polynomial ideal

Introduction

groebner::normalf(p, polys) computes a normal form of the polynomial p by complete reduction modulo all polynomials in the list polys.

Call(s)

groebner::normalf(p, polys <, order>)

Parameters

p - a polynomial or a polynomial expression
polys - a list of polynomials of the same type as p. In particular, polys must be a list of polynomial expressions with rational coefficients, if p is a polynomial expression.
order - one of the identifiers DegInvLexOrder, DegreeOrder, and LexOrder, or a user-defined term ordering of type Dom::MonomOrdering. The default ordering is DegInvLexOrder.

Returns

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

Related Functions

groebner::gbasis, poly

Details

Example 1

We consider the ideal generated by the following polynomials:

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

We compute the normal form of the following polynomial p modulo the ideal generated by p1, p2 with respect to lexicographical ordering:

>> p :=  poly(x^2*y - 2*x*y + 1, [x,y]):
   groebner::normalf(p, [p1, p2], LexOrder);
                              3      2
                    poly(- 2 y  + 2 y  - y + 1, [x, y])

Note that p1, p2 do not form a Gröbner basis. The corresponding Gröbner basis leads to a different normal form of p:

>> groebner::normalf(p, groebner::gbasis([p1, p2]), LexOrder)
                         poly(- 5/9 y + 1, [x, y])
>> delete p1, p2, p:

Background

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