Nitin Saxena

PARAMETERS OF INTEGRAL CIRCULANT GRAPHS AND PERIODIC QUANTUM DYNAMICS

The means for simultaneous scouring of metal surfaces contains a waste product in manufacture of fodder yeast, citric acid, ammonium citrate, aqueous solution of sodium gluconate, sulphonated ricinic oil and an inorganic acid, f.e. sulphuric acid respectively in the following weight ratios: 60 to 95%; 2 to 6%; 0.1 to 10%; 0.0 to 4.0%; 0.0 to 20% and 0.0 to 15%. The waste product from fodder yeast manufacture contains itself 0.06 to 0.1% reducing agents; 0.01 to 0.4% phosphates; 0.2 to 0.4% ammonium sulphate; 0.0 to 0.06% furfurol and 0.0 to 0.1% yeast. The means for simultaneous scouring of metal surface from corrosion products, scale and scoria is used in metallurgy, machine construction, agriculture, energetics and all fields where there are conditions for metal corrosion.

Progress on Polynomial Identity Testing-II

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.

Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

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, over fields of zero or large characteristic. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely: - depth-3 (Ω Π Ω) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and - constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), and brings them under one unifying technique. In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation.

An Almost Optimal Rank Bound for Depth-3 Identities

We show that the rank of a depth-

Deterministic Polynomial Time Algorithms for Matrix Completion Problems

3

Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter

circuit (over any field) that is simple, minimal and zero is at most

Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

O(k^3\log d)

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

. The previous best rank bound known was

Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter

2^{O(k^2)}(\log d)^{k-2}

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank