Joos Heintz

Definability and fast quantifier elimination in algebraically closed fields

Abstract The Bezout-Inequality, an affine version (not including multiplicities) of the classical Bezout-Theorem is derived for applications in algebraic complexity theory. Upper bounds for the cardinality and number of sets definable by first order formulas over algebraically closed fields are given. This is used for fast quantifier elimination in algebraically closed fields.

Real quantifier elimination is doubly exponential

We show that quantifier elimination over real closed fields can require doubly exponential space (and hence time). This is done by explicitly constructing a sequence of expressions whose length is linear in the number of quantifiers, but whose quantifier-free expression has length doubly exponential in the number of quantifiers. The results can be applied to cylindrical algebraic decomposition, showing that this can be doubly exponential. The double exponents of our lower bounds are about one fifth of the double exponents of the best-known upper bounds.

Absolute Primality of Polynomials is Decidable in Random Polynomial Time in the Number of Variables

Let F be a n-variate polynomial with deg F = d over an infinite field k0. Absolute primality of F can be decided randomly in time polynomial in n and exponential in d5 and determinalistically in time exponential in d6 + n2 d3.

Lower bounds for polynomials with algebraic coefficients

We give a method, based on algebraic geometry, to show lower bounds for the complexity of polynomials with algebraic coefficients. Typical examples are polynomials with coefficients which are roots of unity, such as Σj=1de2πiiXi and Σj=ide2πipiXj where pj is the jth prime number. We apply the method also to systems of linear equations.

Commutative Algebras of Minimal Rank

Abstract If A is a finite-dimensional associative k-algebra with unit, then its rank R( A ), i.e. its bilinear complexity, is never less than 2dim A −# M ( A ) . A is said to be of minimal rank if R( A )=2dim A −# M ( A ) . In this paper we determine for infinite perfect fields k all commutative k-algebras of minimal rank. Roughly speaking, these algebras are built up from simply generated structures which annihilate each other. Furthermore, we indicate how this result can be used to obtain new lower bounds for the rank of specific commutative algebras.

Algorithmic aspects of Suslin's proof of Serre's conjecture

AbstractLetF be a unimodularr×s matrix with entries beingn-variate polynomials over an infinite fieldK. Denote by deg(F) the maximum of the degrees of the entries ofF and letd=1+deg(F). We describe an algorithm which computes a unimodulars×s matrixM with deg(M)=(rd)O(n) such thatFM=[Ir,O], where [Ir,O] denotes ther×s matrix obtained by adding to ther×r unit matrixIrs−r zero columns.We present the algorithm as an arithmetic network with inputs fromK, and we count field operations and comparisons as unit cost.The sequential complexity of our algorithm amounts to

Intrinsic complexity estimates in polynomial optimization

S^{O(r^2 )} r^{O(n^2 )} d^{O(n^2 + r^2 )}

On polynomials with symmetric Galois group which are easy to compute

field operations and comparisons inK whereas its parallel complexity isO(n4r4log2(srd)).The complexity bounds and the degree bound for deg(M) mentioned above are optimal in order. Our algorithm is inspired by Suslins proof of Serres Conjecture.