# Bit complexity for multi-homogeneous polynomial system solving - Application to polynomial minimization

@article{Din2018BitCF, title={Bit complexity for multi-homogeneous polynomial system solving - Application to polynomial minimization}, author={Mohab Safey El Din and {\'E}ric Schost}, journal={J. Symb. Comput.}, year={2018}, volume={87}, pages={176-206} }

## 17 Citations

Multilinear polynomial systems: Root isolation and bit complexity

- Mathematics, Computer ScienceJ. Symb. Comput.
- 2021

Towards Mixed Gröbner Basis Algorithms: the Multihomogeneous and Sparse Case

- Computer Science, MathematicsISSAC
- 2018

A dedicated algorithm that exploits different algebraic properties that performs no reduction to zero for mixed, square, and 0-dimensional multihomogeneous polynomial systems is presented.

Real Root Finding for Equivariant Semi-algebraic Systems

- Mathematics, Computer ScienceISSAC
- 2018

The emptiness of basic semi-algebraic sets defined by s polynomials of degree d in time (sn)O(d) can be decided, which improves the state-of-the-art which is exponential in n.

Real root finding for low rank linear matrices

- Computer Science, MathematicsApplicable Algebra in Engineering, Communication and Computing
- 2019

Computer algebra algorithms, based on advanced methods for polynomial system solving, are designed to solve the problem of finding elements of low rank in a real affine subspace of dimension n efficiently and exactly.

Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients

- Computer Science, MathematicsArXiv
- 2021

It is proved that, actually, certificates of non-negativity modulo gradient ideals can be obtained exactly, over the rationals if the polynomial under consideration has rational coefficients and the authors provide exact algorithms to compute them.

Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems

- MathematicsJ. Symb. Comput.
- 2021

Homotopy techniques for solving sparse column support determinantal polynomial systems

- Mathematics, Computer ScienceJ. Complex.
- 2021

On the bit complexity of finding points in connected components of a smooth real hypersurface

- Mathematics, Computer ScienceISSAC
- 2020

The main contribution is a quantitative analysis of several genericity statements, such as Thom's weak transversality theorem or Noether normalization properties for polar varieties, which are proved to generically ensure certain desirable geometric properties.

Exact Algorithms for Semidefinite Programs with Degenerate Feasible Set

- Mathematics, Computer ScienceISSAC
- 2018

This paper designs an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and it is proved that solving such problems can be done in polynomial time if either n or m is fixed.

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