204 research outputs found
Dedekind Zeta Functions and the Complexity of Hilbert's Nullstellensatz
Let HN denote the problem of determining whether a system of multivariate
polynomials with integer coefficients has a complex root. It has long been
known that HN in P implies P=NP and, thanks to recent work of Koiran, it is now
known that the truth of the Generalized Riemann Hypothesis (GRH) yields the
implication that HN not in NP implies P is not equal to NP. We show that the
assumption of GRH in the latter implication can be replaced by either of two
more plausible hypotheses from analytic number theory. The first is an
effective short interval Prime Ideal Theorem with explicit dependence on the
underlying field, while the second can be interpreted as a quantitative
statement on the higher moments of the zeroes of Dedekind zeta functions. In
particular, both assumptions can still hold even if GRH is false. We thus
obtain a new application of Dedekind zero estimates to computational algebraic
geometry. Along the way, we also apply recent explicit algebraic and analytic
estimates, some due to Silberman and Sombra, which may be of independent
interest.Comment: 16 pages, no figures. Paper corresponds to a semi-plenary talk at
FoCM 2002. This version corrects some minor typos and adds an
acknowledgements sectio
Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case
The objective of this paper is to show how the recently proposed method by
Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to
a case of real polynomial equation solving. Our main result concerns the
problem of finding one representative point for each connected component of a
real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a
method for symbolically solving a zero-dimensional polynomial equation system
in the affine (and toric) case. Its main feature is the use of adapted data
structure: Arithmetical networks and straight-line programs. The algorithm
solves any affine zero-dimensional equation system in non-uniform sequential
time that is polynomial in the length of the input description and an
adequately defined {\em affine degree} of the equation system. Replacing the
affine degree of the equation system by a suitably defined {\em real degree} of
certain polar varieties associated to the input equation, which describes the
hypersurface under consideration, and using straight-line program codification
of the input and intermediate results, we obtain a method for the problem
introduced above that is polynomial in the input length and the real degree.Comment: Late
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
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