Some proposals for open ended problems.
Implement a numerical scaling algorithm that tests semistability of projective hypersurfaces. The input is a homogeneous polynomial. The output is either destabilizing one-parameter subgroup, or –ideally– an invariant that does not vanish for the hypersurface.
The Grassmannian Gr(1,P^4) of lines in 4-space is a 6-dimensional variety in P^9. Write its Chow form as the determinant of a 5 × 5-matrix whose entries are linear forms in the 10 choose 6 = 210 brackets, ready to be evaluated in a computer algebra system. Do the same with the Hurwitz form, and pass to higher Grassmannians. Determine the Chow polytopes.
- Let X_r(m,n,p) denote the set of complex tensors in C^m \otimes C^n \otimes C^p with border rank at most r. It can be interpreted as the affine cone over a higher secant variety of the Segre variety P(C^m) x P(C^n) x P(C^p). The dimension of X_r(m,n,p) is known in many cases, even though a full picture is still lacking. What can we say about the degree of X_r(m,n,p), when seen as a projective variety? The case m=1 is known. One should first look at special formats, e.g. m=2. Can one make asymptotic statements? It is possible to compute this degree in small cases, e.g., using homotopy continuation with julia?
- $$\sum_{i=0}^\infty x^i$$
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