Sébastien Tavenas

Improved bounds for reduction to depth 4 and depth 3

Koiran showed that if an n-variate polynomial f n of degree d (with d = n O ( 1 ) ) is computed by a circuit of size s, then it is also computed by a homogeneous circuit of depth four and of size 2 O ( d log ? ( n ) log ? ( s ) ) . Using this result, Gupta, Kamath, Kayal and Saptharishi found an upper bound for the size of a depth three circuit computing f n .We improve here Koirans bound. Indeed, we transform an arithmetic circuit into a depth four circuit of size 2 ( O ( d log ? ( d s ) log ? ( n ) ) ) . Then, mimicking the proof in [2], it also implies a 2 ( O ( d log ? ( d s ) log ? ( n ) ) ) upper bound for depth three circuits.This new bound is almost optimal since a 2 ? ( d ) lower bound is known for the size of homogeneous depth four circuits such that gates at the bottom have fan-in at most d . Finally, we show that this last lower bound also holds if the fan-in is at least d .

Improved Bounds for Reduction to Depth 4 and Depth 3

Koiran [7] showed that if an n-variate polynomial of degree d (with d = n O(1)) is computed by a circuit of size s, then it is also computed by a homogeneous circuit of depth four and of size \(2^{O(\sqrt{d}\log(d)\log(s))}\). Using this result, Gupta, Kamath, Kayal and Saptharishi [6] gave an exp \(\left(O\left(\sqrt{d\log(d)\log(n)\log(s)}\right)\right)\) upper bound for the size of the smallest depth three circuit computing an n-variate polynomial of degree d = n O(1) given by a circuit of size s.

A Wronskian approach to the real τ-conjecture

According to the real ?-conjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent.In this paper, we use the Wronksian determinant to give an upper bound on the number of real roots of sums of products of sparse polynomials of a special form. We focus on the case where the number of distinct sparse polynomials is small, but each polynomial may be repeated several times. We also give a deterministic polynomial identity testing algorithm for the same class of polynomials.Our proof techniques are quite versatile; they can in particular be applied to some sparse geometric problems that do not originate from arithmetic circuit complexity. The paper should therefore be of interest to researchers from these two communities (complexity theory and sparse polynomial systems).

On the size of homogeneous and of depth four formulas with low individual degree

Let r be an integer. Let us call a polynomial f as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. Specifically, first define the formal degree of a node a with respect to a variable x inductively as follows. For a leaf it is 1 if a is labelled with x and zero otherwise; for an internal node labelled with * (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits.

Building Efficient and Compact Data Structures for Simplicial Complexes

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the ST while retaining its functionalities. In addition, we propose two new data structures called the Maximal Simplex Tree and the Simplex Array List. We analyze the compressed ST, the Maximal Simplex Tree, and the Simplex Array List under various settings.

An Almost Cubic Lower Bound for Depth Three Arithmetic Circuits

We show an almost cubic lower bound on the size of any depth three arithmetic circuit computing an explicit multilinear polynomial in n variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson [CCC, 1999].

On the Intersection of a Sparse Curve and a Low-Degree Curve: A Polynomial Version of the Lost Theorem

Consider a system of two polynomial equations in two variables:

Log-Concavity and Lower Bounds for Arithmetic Circuits

\begin{aligned} F(X,Y)=G(X,Y)=0, \end{aligned}