The number of real roots of a bivariate polynomial on a line

We prove that a bivariate polynomial f with exactly t non-zero terms, restricted to a real line {y=ax+b}, either has at most 6t-4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether a linear polynomial divides a bivariate polynomial (with exactly t non-zero terms) over a real number field K within [ log(H(f)H(a)H(b)) [K:Q}] log(deg(f)) t]^{O(1)} bit operations.Comment: 6 pages, no figure

Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t>=0. Denote by Bm(t,K) the maximum number of non-zero roots in K, counted with multiplicities, of a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)<=t^2 Bm(t,K) for any local field L with a non-archimedean valuation v such that v(n)=0 for all non-zero integer n and residue field K, and that Bm(t,K)<=(t^2-t+1)(p^f-1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K)>=(2t-1)(p^f-1), which gives the sharp estimation Bm(2,K)=3(p^f-1) for trinomials when p>2+e

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

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