# traceCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)

## Synopsis

• Usage:
numRealTrace(R)
• Inputs:
• S, , an Artinian ring
• f, , a rational univariate polynomial
• I, an ideal, the ideal generated by f
• l, a list, a system of rational univariate polynomials
• Outputs:
• an integer, the number of real points of SpecR

## Description

This computes the number of real points of SpecR, where R is an Artinian ring with characteristic zero

 i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing i2 : F = {y^2 - x^2 - 1,x - y^2 + 4*y - 2} 2 2 2 o2 = {- x + y - 1, - y + x + 4y - 2} o2 : List i3 : I = ideal F 2 2 2 o3 = ideal (- x + y - 1, - y + x + 4y - 2) o3 : Ideal of R i4 : S = R/I o4 = S o4 : QuotientRing i5 : traceCount(S) o5 = 2
 i6 : R = QQ[x,y] o6 = R o6 : PolynomialRing i7 : I = ideal(1 - x^2*y + 2*x*y^2, y - 2*x - x*y + x^2) 2 2 2 o7 = ideal (- x y + 2x*y + 1, x - x*y - 2x + y) o7 : Ideal of R i8 : traceCount(I) o8 = 3

• traceForm -- the trace quadratic form of a rational polynomial in an Artinian ring

## Ways to use traceCount :

• "traceCount(Ideal)"
• "traceCount(List)"
• "traceCount(QuotientRing)"
• "traceCount(RingElement)"

## For the programmer

The object traceCount is .