Factoring bivariate sparse (lacunary) polynomials

We present a deterministic algorithm for computing all irreducible factors of degree $\le d$ of a given bivariate polynomial $f\in K[x,y]$ over an algebraic number field $K$ and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in $d$. Moreover, we show that the factors over \Qbarra of degree $\le d$ which are not binomials can also be computed in time polynomial in the sparse length of the input and in $d$.Comment: 20 pp, Latex 2e. We learned on January 23th, 2006, that a multivariate version of Theorem 1 had independently been achieved by Erich Kaltofen and Pascal Koira

Detecting lacunary perfect powers and computing their roots

We consider solutions to the equation f = h^r for polynomials f and h and integer r > 1. Given a polynomial f in the lacunary (also called sparse or super-sparse) representation, we first show how to determine if f can be written as h^r and, if so, to find such an r. This is a Monte Carlo randomized algorithm whose cost is polynomial in the number of non-zero terms of f and in log(deg f), i.e., polynomial in the size of the lacunary representation, and it works over GF(q)[x] (for large characteristic) as well as Q[x]. We also give two deterministic algorithms to compute the perfect root h given f and r. The first is output-sensitive (based on the sparsity of h) and works only over Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second approach to computing h, which is extremely efficient and works over both GF(q)[x] (for large characteristic) and Q[x], but depends on a number-theoretic conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of these algorithms are unconditionally polynomial-time in the lacunary size of the input polynomial f. Finally, we demonstrate the efficiency of the randomized detection algorithm and the latter perfect root computation algorithm with an implementation in the C++ library NTL.Comment: to appear in Journal of Symbolic Computation (JSC), 201

Near NP-Completeness for Detecting p-adic Rational Roots in One Variable

We show that deciding whether a sparse univariate polynomial has a p-adic rational root can be done in NP for most inputs. We also prove a polynomial-time upper bound for trinomials with suitably generic p-adic Newton polygon. We thus improve the best previous complexity upper bound of EXPTIME. We also prove an unconditional complexity lower bound of NP-hardness with respect to randomized reductions for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss how our results complement analogous results over the real numbers.Comment: 8 pages in 2 column format, 1 illustration. Submitted to a conferenc

New Bounds on Quotient Polynomials with Applications to Exact Divisibility and Divisibility Testing of Sparse Polynomials

A sparse polynomial (also called a lacunary polynomial) is a polynomial that has relatively few terms compared to its degree. The sparse-representation of a polynomial represents the polynomial as a list of its non-zero terms (coefficient-degree pairs). In particular, the degree of a sparse polynomial can be exponential in the sparse-representation size. We prove that for monic polynomials $f, g \in \mathbb{C}[x]$ such that $g$ divides $f$, the $\ell_2$-norm of the quotient polynomial $f/g$ is bounded by $\lVert f \rVert_1 \cdot \tilde{O}(\lVert{g}\rVert_0^3\text{deg}^2{ f})^{\lVert{g}\rVert_0 - 1}$. This improves upon the exponential (in $\text{deg}{ f}$) bounds for general polynomials and implies that the trivial long division algorithm runs in time quasi-linear in the input size and number of terms of the quotient polynomial $f/g$, thus solving a long-standing problem on exact divisibility of sparse polynomials. We also study the problem of bounding the number of terms of $f/g$ in some special cases. When $f, g \in \mathbb{Z}[x]$ and $g$ is a cyclotomic-free (i.e., it has no cyclotomic factors) trinomial, we prove that $\lVert{f/g}\rVert_0 \leq O(\lVert{f}\rVert_0 \text{size}({f})^2 \cdot \log^6{\text{deg}{ g}})$. When $g$ is a binomial with $g(\pm 1) \neq 0$, we prove that the sparsity is at most $O(\lVert{f}\rVert_0 ( \log{\lVert{f}\rVert_0} + \log{\lVert{f}\rVert_{\infty}}))$. Both upper bounds are polynomial in the input-size. We leverage these results and give a polynomial time algorithm for deciding whether a cyclotomic-free trinomial divides a sparse polynomial over the integers. As our last result, we present a polynomial time algorithm for testing divisibility by pentanomials over small finite fields when $\text{deg}{ f} = \tilde{O}(\text{deg}{ g})$

Efficiently Detecting Torsion Points and Subtori

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse encoding), under a plausible assumption on primes in arithmetic progression. In particular, our hypothesis can still hold even under certain failures of the Generalized Riemann Hypothesis, such as the presence of Siegel-Landau zeroes. Furthermore, we give a similar (but UNconditional) complexity upper bound for n=1. Finally, letting T be any algebraic subgroup of (C^*)^n we show that deciding whether X contains T is coNP-complete (relative to an even more efficient encoding),unconditionally. We thus obtain new non-trivial families of multivariate polynomial systems where deciding the existence of complex roots can be done unconditionally in the polynomial hierarchy -- a family of complexity classes lying between PSPACE and P, intimately connected with the P=?NP Problem. We also discuss a connection to Laurent's solution of Chabauty's Conjecture from arithmetic geometry.Comment: 21 pages, no figures. Final version, with additional commentary and references. Also fixes a gap in Theorems 2 (now Theorem 1.3) regarding translated subtor

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation