[BOOK][B] Condition: The geometry of numerical algorithms
P Bürgisser, F Cucker - 2013 - books.google.com
This book gathers threads that have evolved across different mathematical disciplines into
seamless narrative. It deals with condition as a main aspect in the understanding of the …
seamless narrative. It deals with condition as a main aspect in the understanding of the …
On Łojasiewicz inequalities and the effective Putinar's Positivstellensatz
L Baldi, B Mourrain, A Parusiński - Journal of Algebra, 2025 - Elsevier
The representation of positive polynomials on a semi-algebraic set in terms of sums of
squares is a central question in real algebraic geometry, which the Positivstellensatz …
squares is a central question in real algebraic geometry, which the Positivstellensatz …
Computing the homology of basic semialgebraic sets in weak exponential time
P Bürgisser, F Cucker, P Lairez - Journal of the ACM (JACM), 2018 - dl.acm.org
We describe and analyze an algorithm for computing the homology (Betti numbers and
torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is …
torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is …
Computing the homology of real projective sets
We describe and analyze a numerical algorithm for computing the homology (Betti numbers
and torsion coefficients) of real projective varieties. Here numerical means that the algorithm …
and torsion coefficients) of real projective varieties. Here numerical means that the algorithm …
Computing the homology of semialgebraic sets. I: Lax formulas
P Bürgisser, F Cucker, J Tonelli-Cueto - Foundations of Computational …, 2020 - Springer
We describe and analyze an algorithm for computing the homology (Betti numbers and
torsion coefficients) of closed semialgebraic sets given by Boolean formulas without …
torsion coefficients) of closed semialgebraic sets given by Boolean formulas without …
Plantinga-Vegter algorithm takes average polynomial time
F Cucker, AA Ergür, J Tonelli-Cueto - Proceedings of the 2019 on …, 2019 - dl.acm.org
We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga
and Vegter. The first complexity analysis of the\pv~ Algorithm is due to Burr, Gao and …
and Vegter. The first complexity analysis of the\pv~ Algorithm is due to Burr, Gao and …
Probabilistic condition number estimates for real polynomial systems I: A broader family of distributions
We consider the sensitivity of real roots of polynomial systems with respect to perturbations
of the coefficients. In particular—for a version of the condition number defined by Cucker and …
of the coefficients. In particular—for a version of the condition number defined by Cucker and …
On continued fraction expansion of real roots of polynomial systems, complexity and condition numbers
We elaborate on a correspondence between the coefficients of a multivariate polynomial
represented in the Bernstein basis and in a tensor-monomial basis, which leads to …
represented in the Bernstein basis and in a tensor-monomial basis, which leads to …
A numerical algorithm for zero counting. III: Randomization and condition
F Cucker, T Krick, G Malajovich, M Wschebor - Advances in Applied …, 2012 - Elsevier
In a recent paper (Cucker et al., 2008 [8]) we analyzed a numerical algorithm for computing
the number of real zeros of a polynomial system. The analysis relied on a condition number …
the number of real zeros of a polynomial system. The analysis relied on a condition number …
Functional norms, condition numbers and numerical algorithms in algebraic geometry
F Cucker, AA Ergür, J Tonelli-Cueto - Forum of Mathematics, Sigma, 2022 - cambridge.org
In numerical linear algebra, a well-established practice is to choose a norm that exploits the
structure of the problem at hand to optimise accuracy or computational complexity. In …
structure of the problem at hand to optimise accuracy or computational complexity. In …