Critical Point Computations on Smooth Varieties: Degree and Complexity bounds

Mohab Safey El Din 1 Pierre-Jean Spaenlehauer 2
1 PolSys - Polynomial Systems
LIP6 - Laboratoire d'Informatique de Paris 6, Inria de Paris
2 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Let V ⊂ C n be an equidimensional algebraic set and g be an n-variate polynomial with rational coefficients. Computing the critical points of the map that evaluates g at the points of V is a cornerstone of several algorithms in real algebraic geometry and optimization. Under the assumption that the critical locus is finite and that the projective closure of V is smooth, we provide sharp upper bounds on the degree of the critical locus which depend only on deg(g) and the degrees of the generic polar varieties associated to V. Hence, in some special cases where the degrees of the generic polar varieties do not reach the worst-case bounds, this implies that the number of critical points of the evaluation map of g is less than the currently known degree bounds. We show that, given a lifting fiber of V , a slight variant of an algorithm due to Bank, Giusti, Heintz, Lecerf, Matera and Solernó computes these critical points in time which is quadratic in this bound up to logarithmic factors, linear in the complexity of evaluating the input system and polynomial in the number of variables and the maximum degree of the input polynomials.
Document type :
Conference papers
Complete list of metadatas

Cited literature [36 references]  Display  Hide  Download

https://hal.inria.fr/hal-01312750
Contributor : Pierre-Jean Spaenlehauer <>
Submitted on : Monday, May 9, 2016 - 9:13:19 AM
Last modification on : Saturday, March 30, 2019 - 1:26:25 AM
Long-term archiving on : Wednesday, May 25, 2016 - 7:20:13 AM

Files

main.pdf
Files produced by the author(s)

Licence


Copyright

Identifiers

Citation

Mohab Safey El Din, Pierre-Jean Spaenlehauer. Critical Point Computations on Smooth Varieties: Degree and Complexity bounds. International Symposium on Symbolic and Algebraic Computation (ISSAC), Jul 2016, Waterloo, Canada. pp.183--190, ⟨10.1145/2930889.2930929⟩. ⟨hal-01312750⟩

Share

Metrics

Record views

667

Files downloads

174