# Intrinsic Near Quadratic Complexity Bounds for Real Multivariate Root Counting

@inproceedings{Rojas1998IntrinsicNQ, title={Intrinsic Near Quadratic Complexity Bounds for Real Multivariate Root Counting}, author={J. Maurice Rojas}, booktitle={ESA}, year={1998} }

We give a new algorithm, with three versions, for computing the number of real roots of a system of n polynomial equations in n unknowns. The rst version is of Monte Carlo type and, neglecting logarithmic factors, runs in time quadratic in the average number of complex roots of a closely related system. The other two versions run nearly as fast and progressively remove a measure zero locus of failures present in the rst version. Via a slight simpliication of our algorithm, we can also count…

## 9 Citations

TORIC INTERSECTION THEORY FOR AFFINE ROOT COUNTINGJ

- Mathematics
- 1997

Given any polynomial system with xed monomial term structure, we give explicit formulae for the generic number of roots (over any algebraically closed eld) with speciied coordinate vanishing…

Some Speed-Ups and Speed Limits for Real Algebraic Geometry

- Mathematics, Computer ScienceJ. Complex.
- 2000

An algorithm for approximating the real roots of certain sparse polynomial systems with simple and efficient generalization to certain univariate exponential sums and a new and sharper upper bound on the number of connected components of a semi-algebraic set are given.

ON SOLVING FEWNOMIALS OVER INTERVALS IN FEWNOMIAL TIME

- Mathematics, Computer Science
- 2001

It is shown that in the special case m=3 the authors can approximate within e all the roots of f in the interval (0, R) using just O log(D)log Dlog R e �� arithmetic operations.

Solving Degenerate Sparse Polynomial Systems Faster

- MathematicsJ. Symb. Comput.
- 1999

A fast method to find a point in every irreducible component of the zero set Z of F, a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic is presented.

Algebraic Geometry Over Four Rings and the Frontier to Tractability

- Mathematics
- 2000

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of…

DIMACS Series in Discrete Mathematics and Theoretical Computer Science Erratum to : Lower Bounds and Real Algebraic Geometry

- Computer Science, Mathematics
- 2002

This work describes a method, based on the rational univariate reduction (RUR), for computing roots of systems of multivariate polynomials with rational coefficients that has been used successfully on certain degenerate boundary evaluation problems.

The Exact Rational Univariate Representation and Its Application

- Computer Science, Mathematics
- 2003

A method, based on the rational univariate reduction (RUR), for computing roots of systems of multivariate polynomials with rational coefficients that enables exact algebraic computations with the root coordinates and works even if the underlying set of roots is positive dimensional.

On the complexity of diophantine geometry in low dimensions

- Computer ScienceProceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)
- 1999

It is shown that the following two problems can be solved within PSPACE, and the truth of the Generalized Riemann Hypothesis implies that detecting roots in Q/sup n/ for the polynomial systems in problem (I) can be done via a two-round Arthur-Merlin protocol.

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