# traceForm -- trace matrix of an ideal

## Synopsis

• Usage:
traceForm(I)
• Inputs:
• I, a zero dimensional ideal.
• Outputs:
• S, the trace form of the ideal I. The signature of such a form equals the number of real points in the variety V(I).

## Description

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal(x+3*y^2-2*z, x^2-2*y-z, 3*x-4*y+5*z^2) 2 2 2 o2 = ideal (3y + x - 2z, x - 2y - z, 5z + 3x - 4y) o2 : Ideal of R i3 : T = traceForm(I) o3 = | 8 0 -4 -112/15 -18/5 0 16/5 0 | | 0 0 16/5 24/5 32/5 -4 -112/15 -18/5 | | -4 16/5 122/15 2788/225 24/5 -12/5 -64/15 -112/15 | | -112/15 24/5 2788/225 704/45 208/15 -64/15 -2186/225 -96/25 | | -18/5 32/5 24/5 208/15 238/25 -112/15 -96/25 -16/5 | | 0 -4 -12/5 -64/15 -112/15 0 6/5 16/5 | | 16/5 -112/15 -64/15 -2186/225 -96/25 6/5 316/75 12/5 | | 0 -18/5 -112/15 -96/25 -16/5 16/5 12/5 0 | / R \8 / R \8 o3 : Matrix |------------------------------------------| <--- |------------------------------------------| | 2 2 2 | | 2 2 2 | \(3y + x - 2z, x - 2y - z, 5z + 3x - 4y)/ \(3y + x - 2z, x - 2y - z, 5z + 3x - 4y)/