Lower bounds for diophantine approximations

Abstract

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. ¿From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero-dimensional polynomial equation system. This result represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight-line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton's algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

Polar, bipolar and copolar varieties: Real solving of algebraic varieties with intrinsic complexity

“…In order to treat the smooth unbounded case we introduced in [3,4] the concept of dual polar varieties. In the present paper we put emphasis on the geometrical ideas which together with the Kronecker algorithm [16,18,19,24], that solves polynomial equation systems over the complex numbers, lead to our complexity statements.…”

“…(compare for the case p := 1 the estimates (16) and (17) given in [6], Section 4.2). For the rest of the paper we fix a family {σ 1 , .…”

“…We used them for the design of a pseudo-polynomial computer procedure with an intrinsic complexity bound which finds for a given complete intersection variety S with a smooth compact real trace S R algebraic sample points for each connected component of S R if there are such points ( [1,2]). Actually the geometric resolution of polar varieties, thanks to the algoritm Kronecker developed by the TERA-group [16,18,19,24], led directly to a good pseudo-polynomial complexity. Then we dropped successively the hypothesis on compactness of S R (leading to dual polar varieties [3,4]) and eventually the hypothesis on smoothness of S R .…”

Point searching in real singularcomplete intersection varieties: algorithms of intrinsic complexity

“…The procedure Π i is based on the original paradigm [21,20] of a procedure with intrinsic complexity that solves polynomial equation systems over the complex numbers (see also [19,23,17]). …”

“…For this purpose we shall freely refer to terminology, mathematical results and subroutines of [23], where the first streamlined version of the polynomial equation solver [21,20] was presented and implemented as "Kronecker algorithm" (compare also [27]). …”