Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact ... more >>>

Blackbox polynomial identity testing (PIT) affords 'extreme variable-bootstrapping' (Agrawal et al, STOC'18; PNAS'19; Guo et al, FOCS'19). This motivates us to study log-variate read-once oblivious algebraic branching programs (ROABP). We restrict width of ROABP to a constant and study the more general sum-of-ROABPs model. We give the first poly($s$)-time blackbox ... more >>>

We consider the univariate polynomial $f_d:=(x+1)^d$ when represented as a sum of constant-powers of univariate polynomials. We define a natural measure for the model, the support-union, and conjecture that it is $\Omega(d)$ for $f_d$.

We show a stunning connection of the conjecture to the two main problems in algebraic ... more >>>

Finding an irreducible factor, of a polynomial $f(x)$ modulo a prime $p$, is not known to be in deterministic polynomial time. Though there is such a classical algorithm that {\em counts} the number of irreducible factors of $f\bmod p$. We can ask the same question modulo prime-powers $p^k$. The irreducible ... more >>>

Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod p^2$ is irreducible, but $x^2+px \bmod p^2$ has exponentially many factors! We present the first randomized poly($\deg ... more >>>

Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC'18) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few ... more >>>

We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One only need to consider size-$s$ degree-$s$ circuits that depend on the first $\log^{\circ c} s$ variables (where $c$ is a constant and we are ... more >>>

Testing whether a set $\mathbf{f}$ of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is NP$^{\#\rm P}$ (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). ... more >>>

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the ... more >>>

Research in the last decade has shown that to prove lower bounds or to derandomize polynomial identity testing (PIT) for general arithmetic circuits it suffices to solve these questions for restricted circuits. In this work, we study the smallest possibly restricted class of circuits, in particular depth-$4$ circuits, which would ... more >>>

Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\F_q$ of characteristic $p$ (equivalently, constructing the bigger field $\F_{q^{r^e}}$). Both these problems have famous randomized ... more >>>

We give improved hitting-sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known variable order. The best hitting-set known for this case had cost $(nw)^{O(\log n)}$, where $n$ is the number of variables and $w$ is the width of ... more >>>

A read once ABP is an arithmetic branching program with each variable occurring in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity, i.e. ... more >>>

We give a $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious algebraic branching programs (ROABP). The best time-complexity known for this class was $n^{O(\log^2 n)}$ due to Forbes-Saptharishi-Shpilka (STOC 2014), and that too only for multilinear ROABP. We get rid of their ... more >>>

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.

The depth-$3$ model has recently gained much importance, as it has become a stepping-stone to understanding general arithmetic circuits. Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper we take a step towards designing such hitting-sets. We define a notion of ... more >>>

We call a depth-$4$ formula $C$ $\textit{ set-depth-4}$ if there exists a (unknown) partition $X_1\sqcup\cdots\sqcup X_d$ of the variable indices $[n]$ that the top product layer respects, i.e. $C(\mathbf{x})=\sum_{i=1}^k {\prod_{j=1}^{d} {f_{i,j}(\mathbf{x}_{X_j})}}$ $ ,$ where $f_{i,j}$ is a $\textit{sparse}$ polynomial in $\mathbb{F}[\mathbf{x}_{X_j}]$. Extending this definition to any depth - we call ... more >>>

The problem of finding a nontrivial factor of a polynomial $f(x)$ over a finite field $\mathbb{F}_q$ has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the ... more >>>

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic $p>0$, there is no analogous characterization known. In this paper we give the first such criterion. Essentially, it boils down to ... more >>>

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied ... more >>>

Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials $\{f_1,\ldots, f_m\} \subset \mathbb{F}[x_1,\ldots, x_n]$ are called algebraically independent if there is no non-zero polynomial $F$ such that $F(f_1, \ldots, f_m) = 0$. The transcendence degree, $\mbox{trdeg}\{f_1,\ldots, f_m\}$, is the maximal ... more >>>

Finding an efficient solution to the general problem of polynomial identity testing (PIT) is a challenging task. In this work, we study the complexity of two special but natural cases of identity testing - first is a case of depth-$3$ PIT, the other of depth-$4$ PIT.

Our first problem is ... more >>>

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called sps(k,d,n) circuits) over base field F.
It is a major open problem to design a deterministic polynomial time blackbox algorithm
that tests if C is identically zero.
Klivans & Spielman (STOC 2001) observed ... more >>>

We study the problem of identity testing for depth-3 circuits, over the
field of reals, of top fanin k and degree d (called sps(k,d)
identities). We give a new structure theorem for such identities and improve
the known deterministic d^{k^k}-time black-box identity test (Kayal &
Saraf, FOCS 2009) to one ... more >>>

Polynomial identity testing (PIT) is the problem of checking whether a given
arithmetic circuit is the zero circuit. PIT ranks as one of the most important
open problems in the intersection of algebra and computational complexity. In the last
few years, there has been an impressive progress on this ... more >>>

We present new deterministic algorithms for several cases of the maximum rank matrix completion
problem (for short matrix completion), i.e. the problem of assigning values to the variables in
a given symbolic matrix as to maximize the resulting matrix rank. Matrix completion belongs to
the fundamental problems in computational complexity ... more >>>

We study the problem of polynomial identity testing (PIT) for depth
2 arithmetic circuits over matrix algebra. We show that identity
testing of depth 3 (Sigma-Pi-Sigma) arithmetic circuits over a field
F is polynomial time equivalent to identity testing of depth 2
(Pi-Sigma) arithmetic circuits over U_2(F), the ... more >>>

We show that the rank of a depth-3 circuit (over any field) that is simple,
minimal and zero is at most O(k^3\log d). The previous best rank bound known was
2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005).
This almost resolves the rank question first posed by ... more >>>

In this paper we develop techniques that eliminate the need of the Generalized
Riemann Hypothesis (GRH) from various (almost all) known results about deterministic
polynomial factoring over finite fields. Our main result shows that given a
polynomial f(x) of degree n over a finite field k, we ... more >>>

In this work we relate the deterministic
complexity of factoring polynomials (over
finite
fields) to certain combinatorial objects we
call
m-schemes. We extend the known conditional
deterministic subexponential time polynomial
factoring algorithm for finite fields to get an
underlying m-scheme. We demonstrate ... more >>>

In this paper we give the first deterministic polynomial time algorithm for testing whether a {\em diagonal} depth-$3$ circuit $C(\arg{x}{n})$ (i.e. $C$ is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only ... more >>>

We study the identity testing problem for depth $3$ arithmetic circuits ($\Sigma\Pi\Sigma$ circuits). We give the first deterministic polynomial time identity test for $\Sigma\Pi\Sigma$ circuits with bounded top fanin. We also show that the {\em rank} of a minimal and simple $\Sigma\Pi\Sigma$ circuit with bounded top fanin, computing zero, can ... more >>>

We study the complexity of the isomorphism and automorphism problems for finite rings with unity.

We show that both integer factorization and graph isomorphism reduce to the problem of counting
automorphisms of rings. The problem is shown to be in the complexity class $\AM \cap co\AM$
and hence ... more >>>