238 research outputs found
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
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
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