# eliminate

## Synopsis

• Usage:
eliminate(v,J)
• Inputs:
• v, or a list, a variable or list of variables of a polynomial ring R
• J, an ideal, in the ring R
• Outputs:
• an ideal, generated by the elements of J not involving the variables v

## Description

If the ideal J is homogeneous, then an effort is made to use the Hilbert function to speed up the computation.
 i1 : R = ZZ/101[x,a,b,c,d] o1 = R o1 : PolynomialRing i2 : f = x^2+a*x+b 2 o2 = x + x*a + b o2 : R i3 : g = x^2+c*x+d 2 o3 = x + x*c + d o3 : R i4 : time eliminate(x,ideal(f,g)) -- used 0.00368559 seconds 2 2 2 2 o4 = ideal(a*b*c - b*c - a d + a*c*d - b + 2b*d - d ) o4 : Ideal of R i5 : time ideal resultant(f,g,x) -- used 0.00136556 seconds 2 2 2 2 o5 = ideal(- a*b*c + b*c + a d - a*c*d + b - 2b*d + d ) o5 : Ideal of R i6 : sylvesterMatrix(f,g,x) o6 = {-3} | 1 a b 0 | {-2} | 0 1 a b | {-3} | 1 c d 0 | {-2} | 0 1 c d | 4 4 o6 : Matrix R <--- R i7 : discriminant(f,x) 2 o7 = - a + 4b o7 : R

One may also switch the order of arguments: the ideal being the first argument. This usage has been deprecated, and should no longer be used.

## Caveat

The ring R should not be a quotient ring, or a non-commutative ring. Additionally, it would be nice to be able to use a DegreeLimit, or to be able to interrupt the computation.