75 research outputs found
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 of a given bivariate polynomial over an algebraic
number field and their multiplicities, whose running time is polynomial in
the bit length of the sparse encoding of the input and in . Moreover, we
show that the factors over \Qbarra of degree which are not binomials
can also be computed in time polynomial in the sparse length of the input and
in .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
Multivariate Subresultants in Roots
We give rational expressions for the subresultants of n+1 generic polynomials
f_1,..., f_{n+1} in n variables as a function of the coordinates of the common
roots of f_1,..., f_n and their evaluation in f_{n+1}. We present a simple
technique to prove our results, giving new proofs and generalizing the
classical Poisson product formula for the projective resultant, as well as the
expressions of Hong for univariate subresultants in roots.Comment: 22 pages, no figures, elsart style, revised version of the paper
presented in MEGA 2005, accepted for publication in Journal of Algebr
Subresultants in Multiple Roots
We extend our previous work on Poisson-like formulas for subresultants in
roots to the case of polynomials with multiple roots in both the univariate and
multivariate case, and also explore some closed formulas in roots for
univariate polynomials in this multiple roots setting.Comment: 21 pages, latex file. Revised version accepted for publication in
Linear Algebra and its Application
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