The computational complexity of the Chow form

Flattenings and Koszul Young flattenings arising in complexity theory

I find new equations for Chow varieties, their secant varieties, and an additional variety that arises in the study of depth 5 circuits by flattenings and Koszul Young flattenings. This enables a new lower bound for symmetric border rank of $x_1x_2\cdots x_d$ when $d$ is odd, and a lower bound on the size of depth 5 circuits that compute the permanent.

On the bit complexity of polynomial system solving

We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the Bézout number of the system and linear in its bit size. Our algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a "lucky" prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis …

Kähler-Einstein metrics on symmetric general arrangement varieties

We calculate Chow quotients of some families of symmetric \(T\)-varieties. In complexity two we obtain new examples of Kähler-Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional application we determine the homeomorpism class of the orbit space of the compact torus action.

The Hurwitz Form of a Projective Variety

The Hurwitz form of a variety is the discriminant that characterizes linear spaces of complementary dimension which intersect the variety in fewer than degree many points. We study computational aspects of the Hurwitz form, relate this to the dual variety and Chow form, and show why reduced degenerations are special on the Hurwitz polytope.

多项式子代数的计算复杂性

Computational Complexity of Polynomial Subalgebras

本文研究多项式子代数成员资格问题与次数界的计算复杂性，证明其EXPSPACE完全性及齐次情形的PSPACE完全性，并与理想情形及单项式代数进行比较。

On geometric complexity theory: Multiplicity obstructions are stronger than occurrence obstructions

Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. The papers also conjecture that the vanishing behavior of these multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). Thi…

Computational complexity of learning algebraic varieties

We analyze the complexity of fitting a variety, coming from a class of varieties, to a configuration of points in $\Bbb C^n$. The complexity measure, called the algebraic complexity, computes the Euclidean Distance Degree (EDdegree) of a certain variety called the hypothesis variety as the number of points in the configuration increases. For the problem of fitting an $(n-1)$-sphere to a configuration of $m$ points in $\Bbb C^n$, we give a closed formula of the algebraic complexity of the hypothesis variety as $m$ grows for the case of $n=1$. For the case $n>1$ we conjecture a generalization of…