Shallow Circuits with High-Powered Inputs

A polynomial identity testing algorithm must determine whether an input polynomial (given for instance by an arithmetic circuit) is identically equal to 0. In this paper, we show that a deterministic black-box identity testing algorithm for (high-degree) univariate polynomials would imply a lower bound on the arithmetic complexity of the permanent. The lower bounds that are known to follow from derandomization of (low-degree) multivariate identity testing are weaker. To obtain our lower bound it would be sufficient to derandomize identity testing for polynomials of a very specific norm: sums of products of sparse polynomials with sparse coefficients. This observation leads to new versions of the Shub-Smale tau-conjecture on integer roots of univariate polynomials. In particular, we show that a lower bound for the permanent would follow if one could give a good enough bound on the number of real roots of sums of products of sparse polynomials (Descartes' rule of signs gives such a bound for sparse polynomials and products thereof). In this third version of our paper we show that the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots. This is a consequence of a new result on reduction to depth 4 for arithmetic circuits which we establish in a companion paper. We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte

Log-concavity and lower bounds for arithmetic circuits

One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \sum\_{i = 0}^d a\_i X^i \in \mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition a\_i^2 \textgreater{} \tau a\_{i-1}a\_{i+1} for every $i \in \{1,\ldots,d-1\},$ where \tau \textgreater{} 0. Whenever $f$ can be written under the form $f = \sum\_{i = 1}^k \prod\_{j = 1}^m f\_{i,j}$ where the polynomials $f\_{i,j}$ have at most $t$ monomials, it is clear that $d \leq k t^m$. Assuming that the $f\_{i,j}$ have only non-negative coefficients, we improve this degree bound to $d = \mathcal O(k m^{2/3} t^{2m/3} {\rm log^{2/3}}(kt))$ if \tau \textgreater{} 1, and to $d \leq kmt$ if $\tau = d^{2d}$. This investigation has a complexity-theoretic motivation: we show that a suitable strengthening of the above results would imply a separation of the algebraic complexity classes VP and VNP. As they currently stand, these results are strong enough to provide a new example of a family of polynomials in VNP which cannot be computed by monotone arithmetic circuits of polynomial size

The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real {\tau}-conjecture" which is inspired by this connection. The real {\tau}-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real {\tau}-conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.Comment: 16 page

Arithmetic circuits: the chasm at depth four gets wider

In their paper on the "chasm at depth four", Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n*n matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of polynomial size. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also give an application to boolean circuit complexity, and a simple (but suboptimal) reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree

On Σ A Σ A Σ Circuits: The Role of Middle Σ Fan-in, Homogeneity and Bottom Degree.

We study polynomials computed by depth five Σ ∧ Σ ∧ Σ arithmetic circuits where ‘Σ’ and ‘∧’ represent gates that compute sum and power of their inputs respectively. Such circuits compute polynomials of the form Pt i=1 Q αi i , where Qi = Pri j=1 ` dij ij where `ij are linear forms and ri, αi, t > 0. These circuits are a natural generalization of the well known class of Σ ∧ Σ circuits and received significant attention recently. We prove an exponential lower bound for the monomial x1 · · · xn against depth five Σ ∧ Σ [≤n] ∧ [≥21] Σ and Σ ∧ Σ [≤2 √n/1000] ∧ [≥ √n] Σ arithmetic circuits where the bottom Σ gate is homogeneous. Our results show that the fan-in of the middle Σ gates, the degree of the bottom powering gates and the homogeneity at the bottom Σ gates play a crucial role in the computational power of Σ ∧ Σ ∧ Σ circuits

How many zeros of a random sparse polynomial are real?

We investigate the number of real zeros of a univariate $k$-sparse polynomial $f$ over the reals, when the coefficients of $f$ come from independent standard normal distributions. Recently B\"urgisser, Erg\"ur and Tonelli-Cueto showed that the expected number of real zeros of $f$ in such cases is bounded by $O(\sqrt{k} \log k)$. In this work, we improve the bound to $O(\sqrt{k})$ and also show that this bound is tight by constructing a family of sparse support whose expected number of real zeros is lower bounded by $\Omega(\sqrt{k})$. Our main technique is an alternative formulation of the Kac integral by Edelman-Kostlan which allows us to bound the expected number of zeros of $f$ in terms of the expected number of zeros of polynomials of lower sparsity. Using our technique, we also recover the $O(\log n)$ bound on the expected number of real zeros of a dense polynomial of degree $n$ with coefficients coming from independent standard normal distributions

Real zeros of mixed random fewnomial systems

Consider a system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $f_i$ has a prescribed set of exponent vectors described by a set $A_i \subseteq \mathbb{Z}^n$ of cardinality $t_i$, whose convex hull is denoted $P_i$. Assuming that the coefficients of the $f_i$ are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most $(2\pi)^{-\frac{n}{2}} V_0 (t_1-1)\ldots (t_n-1)$. Here $V_0$ denotes the number of vertices of the Minkowski sum $P_1+\ldots + P_n$. However, this bound does not improve over the bound in B\"urgisser et al. (SIAM J. Appl. Algebra Geom. 3(4), 2019) for the unmixed case, where all supports $A_i$ are equal. All arguments equally work for real exponent vectors.Comment: 10 pages. Fixed an error in the interpretation of the old Theorem 1.3, which was hence downgraded to Proposition 1.3. Added a reference, put some minor clarifications and fixed some typos. Converted to ACM two column styl