20 research outputs found
Solving Degenerate Sparse Polynomial Systems Faster
Consider a system F of n polynomial equations in n unknowns, over an
algebraically closed field of arbitrary characteristic. We present a fast
method to find a point in every irreducible component of the zero set Z of F.
Our techniques allow us to sharpen and lower prior complexity bounds for this
problem by fully taking into account the monomial term structure. As a
corollary of our development we also obtain new explicit formulae for the exact
number of isolated roots of F and the intersection multiplicity of the
positive-dimensional part of Z. Finally, we present a combinatorial
construction of non-degenerate polynomial systems, with specified monomial term
structure and maximally many isolated roots, which may be of independent
interest.Comment: This is the final journal version of math.AG/9702222 (``Toric
Generalized Characteristic Polynomials''). This final version is a major
revision with several new theorems, examples, and references. The prior
results are also significantly improve
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from an unpublished result of Steve Smale.Comment: This is the final journal version which will appear in Journal of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
A Geometric Effective Nullstellensatz
We present in this paper a geometric theorem which clarifies and extends in
several directions work of Brownawell, Kollar and others on the effective
Nullstellensatz. To begin with, we work on an arbitrary smooth complex
projective variety X, with previous results corresponding to the case when X is
projective space. In this setting we prove a local effective Nullstellensatz
for ideal sheaves, and a corresponding global division theorem for adjoint-type
bundles. We also make explicit the connection with the intersection theory of
Fulton and MacPherson. Finally, constructions involving products of prime
ideals that appear in earlier work are replaced by geometrically more natural
conditions involving order of vanishing along subvarieties. The main technical
inputs are vanishing theorems, which are used to give a simple
algebro-geometric proof of a theorem of Skoda type, which may be of independent
interest.Comment: Introduction expanded, examples added, work of Sombra discusse
A perturbed differential resultant based implicitization algorithm for linear DPPEs
Let \bbK be an ordinary differential field with derivation . Let
\cP be a system of linear differential polynomial parametric equations in
differential parameters with implicit ideal \id. Given a nonzero linear
differential polynomial in \id we give necessary and sufficient
conditions on for \cP to be dimensional. We prove the existence of
a linear perturbation \cP_{\phi} of \cP so that the linear complete
differential resultant \dcres_{\phi} associated to \cP_{\phi} is nonzero. A
nonzero linear differential polynomial in \id is obtained from the lowest
degree term of \dcres_{\phi} and used to provide an implicitization algorithm
for \cP
Computational Arithmetic Geometry I: Sentences Nearly in the Polynomial Hierarchy
We consider the average-case complexity of some otherwise undecidable or open
Diophantine problems. More precisely, consider the following: (I) Given a
polynomial f in Z[v,x,y], decide the sentence \exists v \forall x \exists y
f(v,x,y)=0, with all three quantifiers ranging over N (or Z). (II) Given
polynomials f_1,...,f_m in Z[x_1,...,x_n] with m>=n, decide if there is a
rational solution to f_1=...=f_m=0. We show that, for almost all inputs,
problem (I) can be done within coNP. The decidability of problem (I), over N
and Z, was previously unknown. We also show that the Generalized Riemann
Hypothesis (GRH) implies that, for almost all inputs, problem (II) can be done
via within the complexity class PP^{NP^NP}, i.e., within the third level of the
polynomial hierarchy. The decidability of problem (II), even in the case m=n=2,
remains open in general.
Along the way, we prove results relating polynomial system solving over C, Q,
and Z/pZ. We also prove a result on Galois groups associated to sparse
polynomial systems which may be of independent interest. A practical
observation is that the aforementioned Diophantine problems should perhaps be
avoided in the construction of crypto-systems.Comment: Slight revision of final journal version of an extended abstract
which appeared in STOC 1999. This version includes significant corrections
and improvements to various asymptotic bounds. Needs cjour.cls to compil
Explicit formulas for the multivariate resultant
We present formulas for the homogenous multivariate resultant as a quotient
of two determinants. They extend classical Macaulay formulas, and involve
matrices of considerably smaller size, whose non zero entries include
coefficients of the given polynomials and coefficients of their Bezoutian.
These formulas can also be viewed as an explicit computation of the morphisms
and the determinant of a resultant complex.Comment: 30 pages, Late