Lower bounds for arithmetic problems

Lower Bounds for Coprimeness and Other Decision Problems in Arithmetic

This talk was about some joint work with Lou van den Dries, in which we try to derive lower bounds for the worst-case, time complexity of functions and decision problems in arithmetic which apply to all—or, in any case, to as many as possible— algorithms. The relevant papers are listed in the bibliography and [3] gives a brief account of how we came to these questions, as well as a fairly compl...

Lower Bounds for Covering Problems

Some Lower Bounds for Estrada Index

Arithmetic Circuit Lower Bounds via MaxRank

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : • As our main result, we prove that any homogeneous d...

Lower Bounds for Dynamic Algebraic Problems

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form “change input xi to value v” or “what is the value of output yi?”. We present techniques for showing lower bounds on ...