# regularRepresentation -- the regular representation of a rational polynomial

## Synopsis

• Usage:
regularRepresentation(f)
regularRepresentation(g,I)
• Inputs:
• f, , an element of an Artinian ring
• g, , a rational polynomial
• I, an ideal, a zero-dimensional ideal in the same ring as g
• Outputs:
• , the standard basis of ring f (resp. (ring g)/I)
• , the matrix of the linear map defined by multiplication by f (resp. g) in ring f (resp. (ring g)/I)

## Description

This command gives the matrix of the linear map defined by multiplication by f (resp. g) in terms of the standard basis of the finite-dimensional vector space ring f (resp. (ring g)/I).

 i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing i2 : I = ideal(y^2 - x^2 - 1,x - y^2 + 4*y - 2) 2 2 2 o2 = ideal (- x + y - 1, - y + x + 4y - 2) o2 : Ideal of R i3 : regularRepresentation(y,I) o3 = (| 1 x xy y |, | 0 0 -3 -2 |) | 0 0 -1 1 | | 0 1 4 0 | | 1 0 4 4 | o3 : Sequence i4 : S = R/I o4 = S o4 : QuotientRing i5 : regularRepresentation(y) o5 = (| 1 x xy y |, | 0 0 -3 -2 |) | 0 0 -1 1 | | 0 1 4 0 | | 1 0 4 4 | o5 : Sequence

## Ways to use regularRepresentation :

• "regularRepresentation(RingElement)"
• "regularRepresentation(RingElement,Ideal)"

## For the programmer

The object regularRepresentation is .