An arithmetic read-once formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of read-once formulas. the following are some of the results that we obtain. 1. Given k ROFs in n variables, over a field F, we give a deterministic (non black-box) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k2). 2. We give an n O(d+k2) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k2) for the sum of k depth d ROFs. If |F | is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs. 3. We give a hitting set of size exp ( Õ( √ n + k 2)) for the sum of k ROFs (without depth restrictions). In particular this implies a sub-exponential time deterministic algorithm for

We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth 1 d and product-depth d + 1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that • There exists a multilinear circuit of product-depth d + 1 and of polynomial size computing f. • Every multilinear circuit of product-depth d computing f has super-polynomial size. 1

We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients ’ vector of a polynomial and the coefficients ’ vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits. Monotone Circuits: We prove a tight 2 Ω(n) lower bound for the size of monotone arithmetic circuits. The highest previous lower bound was 2 Ω( √ n). Orthogonal Formulas: We prove a tight 2 Ω(n) lower bound for the size of orthogonal multilinear formulas (defined, motivated, and studied by Aaronson). Previously, nontrivial lower bounds were only known for subclasses of orthogonal multilinear formulas. Non-Cancelling Formulas: We define and study the new model of non-cancelling multilinear formulas. Roughly speaking, in this model one is not allowed to sum two polynomials that almost cancel each other. The non-cancelling multilinear model is a generalization of both the monotone model and the orthogonal model. We prove lower bounds of n Ω(1) for the depth of non-cancelling multilinear formulas.

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In their seminal paper, Valiant, Skyum, Berkowitz and Rackoff proved that arithmetic circuits can be balanced [VSBR]. That is, [VSBR] showed that for every arithmetic circuit Φ of size s and degree r, there exists an arithmetic circuit Ψ of size poly(r, s) and depth O(log(r) log(s)) computing the same polynomial. In the first part of this paper, we follow the proof of [VSBR] and show that syntactically multilinear arithmetic circuits can be balanced. That is, we show that if Φ is syntactically multilinear, then so is Ψ. Recently, [R04b] proved a super-polynomial separation between multilinear arithmetic formula and circuit size. In the second part of this paper, we use the result of the first part to simplify the proof of this separation. That is, we construct a (simpler) polynomial f(x1,..., xn) such that • Every multilinear arithmetic formula computing f is of size n Ω(log(n)). • There exists a syntactically multilinear arithmetic circuit of size poly(n) and depth O(log 2 (n)) computing f. 1

A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we

We show that any explicit example for a tensor A: [n] r → F with tensor-rank ≥ nr·(1−o(1)) , (where r = r(n) ≤ log n / log log n), implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply superpolynomial lower bounds for the size of general arithmetic formulas. One component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas, that gives the following results: We show that for any n-variate homogenous polynomial f of degree r, if there exists a (fanin-2) ( formula of size s and depth d for f then there exists a homogenous (d+r+1)) formula of size O r · s for f. In particular, for any r ≤ log n, if there exists a polynomial size formula for f then there exists a polynomial size homogenous formula for f. This refutes a conjecture of Nisan and Wigderson [NW95] and shows that superpolynomial lower bounds for homogenous formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas. We show that for any n-variate set-multilinear polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a set-multilinear formula of size O ((d + 2) r · s) for f. In particular, for any r ≤ log n / log log n, if there exists a polynomial size formula for f then there exists a polynomial size set-multilinear formula for f. This shows that super-polynomial lower bounds for set-multilinear formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas.

My main area of research is theoretical computer science, with a focus on algebraic complexity theory. I am also interested in other fields of mathematics, such as probability theory and combinatorics.

Abstract. We show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.