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

  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.},
Multilinear polynomial systems: Root isolation and bit complexity
On the bit complexity of polynomial system solving
Towards Mixed Gröbner Basis Algorithms: the Multihomogeneous and Sparse Case
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Real Root Finding for Equivariant Semi-algebraic Systems
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
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Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients
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Homotopy techniques for solving sparse column support determinantal polynomial systems
On the bit complexity of finding points in connected components of a smooth real hypersurface
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
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.


On the Bit Complexity of Solving Bilinear Polynomial Systems
A careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making the algorithms and complexity analysis applicable to the general case.
Sums of Squares, Moment Matrices and Optimization Over Polynomials
This work considers the problem of minimizing a polynomial over a semialgebraic set defined byPolynomial equations and inequalities, which is NP-hard in general and reviews the mathematical tools underlying these properties.
A Gröbner Free Alternative for Polynomial System Solving
A new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials are introduced, and a new codification of the set of solutions of a positive dimensional algebraic variety is given relying on a new global version of Newton's iterator.
When Polynomial Equation Systems Can Be "Solved" Fast?
It is possible to solve any affine or toric zero-dimensional equation system in non-uniform sequential time which is polynomial in the length of the input description and the “geometric degree” of the equation system.
On the complexity of the multivariate resultant
Computing Parametric Geometric Resolutions
  • É. Schost
  • Computer Science, Mathematics
    Applicable Algebra in Engineering, Communication and Computing
  • 2003
This work presents a probabilistic algorithm to compute a parametric resolution of parameters of a polynomial system of n equations in n unknowns, and presents several applications, notably to computa- tions in the Jacobian of hyperelliptic curves and to questions of real geometry.
Deformation Techniques for Sparse Systems
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Global Optimization with Polynomials and the Problem of Moments
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