# numericalImageDegree -- computes the degree of the image of a variety

## Synopsis

• Usage:
numericalImageDegree W
numericalImageDegree(F, I)
• Inputs:
• W, , a pseudo-witness set for $F(V(I))$
• F, a matrix, or list, or ring map, specifying a map
• I, an ideal, which is prime, specifying a source variety $V(I)$
• Optional inputs:
• DoRefinements => ..., default value false, computes a pseudo-witness set for the image of a variety
• DoTraceTest => ..., default value true, computes a pseudo-witness set for the image of a variety
• MaxAttempts => ..., default value 5, computes a pseudo-witness set for the image of a variety
• MaxPoints => ..., default value infinity, computes a pseudo-witness set for the image of a variety
• MaxThreads => ..., default value 1, specifies the maximum number of processor threads
• Repeats => ..., default value 3, computes a pseudo-witness set for the image of a variety
• Software => ..., default value M2engine, specify software for homotopy continuation
• Threshold => ..., default value 5, computes a pseudo-witness set for the image of a variety
• TraceThreshold => ..., default value .00001, computes a pseudo-witness set for the image of a variety
• Verbose => ..., default value true, display detailed output
• Outputs:
• an integer, the degree of $F(V(I))$

## Description

This method computes the degree of the image of a variety, by computing a pseudo-witness set for the image (cf. pseudo-witness set for more on the techniques and options used).

If a pseudo-witness set has already been computed, then to avoid repetitive calculation one may run this function with the pseudo-witness set as input.

The following example determines the degree of the Grassmannian $Gr(2,4)$ of $P^1$'s in $P^3$, under its Pl&uuml;cker embedding in $P^5$.

 i1 : R = CC[x_(1,1)..x_(2,4)]; I = ideal 0_R; o2 : Ideal of R i3 : F = (minors(2, genericMatrix(R, 2, 4)))_*; i4 : numericalImageDegree(F, I, Repeats => 2, Verbose => false) o4 = 2

## Ways to use numericalImageDegree :

• "numericalImageDegree(List,Ideal)"
• "numericalImageDegree(Matrix,Ideal)"
• "numericalImageDegree(PseudoWitnessSet)"
• "numericalImageDegree(RingMap,Ideal)"

## For the programmer

The object numericalImageDegree is .