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 $\text{deg}{ f} = \tilde{O}(\text{deg}{ g})$

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

Some Speed-Ups and Speed Limits for Real Algebraic Geometry

We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel in that it is stated in terms of the volumes of certain polytopes and, for a large class of inputs, beats the best previous bounds by a factor exponential in the number of variables. (2) A new algorithm for approximating the real roots of certain sparse polynomial systems. Two features of our algorithm are (a) arithmetic complexity polylogarithmic in the degree of the underlying complex variety (as opposed to the super-linear dependence in earlier algorithms) and (b) a simple and efficient generalization to certain univariate exponential sums. (3) Detecting whether a real algebraic surface (given as the common zero set of some input straight-line programs) is not smooth can be done in polynomial time within the classical Turing model (resp. BSS model over C) only if P=NP (resp. NP<=BPP). The last result follows easily from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of Complexity. More typos are corrected, and a new section is added where the bounds here are compared to an earlier result of Benedetti, Loeser, and Risler. The LaTeX source needs the ajour.cls macro file to compil