# imageUnderRationalMap -- Compute the image of the scheme under a rational map

## Synopsis

• Usage:
I = imageUnderRationalMap(J,L)
• Inputs:
• J, an ideal, in a polynomial ring
• L, , of homogeneous polynomials of equal degrees
• Outputs:
• I, an ideal, of the image of the scheme defined by J under the rational map defined by L

## Description

 i1 : setRandomSeed("alpha"); i2 : p=nextPrime 10000 o2 = 10007 i3 : kk=ZZ/p o3 = kk o3 : QuotientRing i4 : R=kk[t_0,t_1] o4 = R o4 : PolynomialRing i5 : I=ideal 0_R o5 = ideal 0 o5 : Ideal of R i6 : L=matrix{{t_0^4,t_0^3*t_1,t_0*t_1^3,t_1^4}} o6 = | t_0^4 t_0^3t_1 t_0t_1^3 t_1^4 | 1 4 o6 : Matrix R <--- R i7 : J=imageUnderRationalMap(I,L) 3 2 2 2 3 2 o7 = ideal (x x - x x , x - x x , x x - x x , x - x x ) 1 2 0 3 2 1 3 0 2 1 3 1 0 2 o7 : Ideal of kk[x ..x ] 0 3 i8 : betti J 0 1 o8 = total: 1 4 0: 1 . 1: . 1 2: . 3 o8 : BettiTally

## Ways to use imageUnderRationalMap :

• "imageUnderRationalMap(Ideal,Matrix)"

## For the programmer

The object imageUnderRationalMap is .