hadamardProduct(Ideal,Ideal) -- Hadamard product of two homogeneous ideals

Synopsis

Function: hadamardProduct
Usage:

hadamardProduct(I,J)
Inputs:
- I, an ideal, (homogeneous)
- J, an ideal, (homogeneous)
Outputs:
- an ideal,

Description

Given two projective subvarieties $X$ and $Y$, their Hadamard product is defined as the Zariski closure of the set of (well-defined) entrywise products of pairs of points in the cartesian product $X \times Y$. This can also be regarded as the image of the Segre product of $X \times Y$ via the linear projection on the $z_{ii}$ coordinates. The latter is the way the function is implemented.

Consider for example the entrywise product of two points.

i1 : S = QQ[x,y,z,t];

i2 : p = point {1,1,1,2};

i3 : q = point {1,-1,-1,-1};

i4 : idealOfProjectivePoints({p*q},S)

o4 = ideal (2z - t, 2y - t, 2x + t)

o4 : Ideal of S

This can be computed also from their defining ideals as explained.

i5 : IP = ideal(x-y,x-z,2*x-t)

o5 = ideal (x - y, x - z, 2x - t)

o5 : Ideal of S

i6 : IQ = ideal(x+y,x+z,x+t)

o6 = ideal (x + y, x + z, x + t)

o6 : Ideal of S

i7 : hadamardProduct(IP,IQ)

o7 = ideal (2z - t, 2y - t, 2x + t)

o7 : Ideal of S

We can also consider Hadamard product of higher dimensional varieties. For example, the Hadamard product of two lines.

i8 : I = ideal(random(1,S),random(1,S));

o8 : Ideal of S

i9 : J = ideal(random(1,S),random(1,S));

o9 : Ideal of S

i10 : hadamardProduct(I,J)

                     2                                             
o10 = ideal(97048224x  - 42121625x*y + 287249760x*z - 14791875y*z +
      -----------------------------------------------------------------------
               2
      88905600z  + 4761000x*t + 12420000z*t)

o10 : Ideal of S

Ways to use this method: