Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve

Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas

In this paper we study the identity testing problem of arithmetic read-once formulas (ROF) and some related models. A read-once formula is formula (a circuit whose underlying graph is a tree) in which the operations are {+,x} and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for read-once formulas, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of [Shpilka-Volkovich, 2015

Invariant Theory, Tensors and Computational Complexity

The main problem addressed in this dissertation is the problem of giving strong upper bounds on the degree of generators for invariant rings. In the cases of matrix invariants and matrix semi-invariants, we give polynomial upper bounds. An exciting consequence of these bounds is a polynomial time algorithm for rational identity testing. We use an approach inspired by ideas from Popov and Derksen to reduce the problem to finding invariants that define the null cone. The theory of blow-ups of matrix spaces and non-commutative rank is crucial in finding invariants that define the null cone. We also give a polynomial time algorithm for deciding if the orbit closures of two points intersect for matrix invariants and semi-invariants. In addition, we give some applications for proving lower bounds on the border rank of tensors.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144049/1/visu_1.pd