Solving parametric systems of polynomial equations over the reals through Hermite matrices

Le, Huu Phuoc; Din, Mohab Safey El

doi:10.1016/j.jsc.2021.12.002

Cited by 8 publications

(11 citation statements)

References 43 publications

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“…The latter is done through the so-called critical point method which consists in computing the critical points of a well-chosen polynomial map reaching its extrema on all connected components of the considered real algebraic set. Such a solving scheme has been refined and improved in particular cases such as the one considered in [25,Section 3. ], where the semi-algebraic set is open and where explicit complexity constants in the big-O exponent are well controlled.…”

Section: Sample Points Algorithmsmentioning

confidence: 99%

“…, x n ] and returns an encoding of at least one point per connected component of the real solution set to the input system. When the input polynomials have degree at most D, this can be done in time singly exponential in n and polynomial in D and s using the critical point method introduced in [20] and developed in [27,2,25]. The algorithm in [25] is the one which we will specifically use.…”

Section: Introductionmentioning

confidence: 99%

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Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry

Chablat,

Prébet,

Din

et al. 2022

Preprint

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Cuspidal robots are robots with at least two inverse kinematic solutions that can be connected by a singularity-free path. Deciding the cuspidality of a generic 3R robots has been studied in the past, but extending the study to six-degree-of-freedom robots can be a challenging problem. Many robots can be modeled as a polynomial map together with a real algebraic set so that the notion of cuspidality can be extended to these data.In this paper we design an algorithm that, on input a polynomial map in n indeterminates, and s polynomials in the same indeterminates describing a real algebraic set of dimension d, decides the cuspidality of the restriction of the map to the real algebraic set under consideration. Moreover, if D and τ are respectively the maximum degree and a bound on the bit size of the coefficients of the input polynomials, this algorithm runs in time log-linear in τ and polynomial in ((s + d)nD) O(n 2 ) .It relies on many high-level algorithms in computer algebra which use advanced methods on real algebraic sets and critical loci of polynomial maps. As far as we know, this is the first algorithm that tackles the cuspidality problem from a general point of view.

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Section: Sample Points Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry

Chablat,

Prébet,

Din

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…generated by F is radical and zero-dimensional and computes a quantifier-free formula Φ F in y such that Z(Φ F ) is dense in the interior of π(V (F ) ∩ R n+t ). For this task, we refer to the algorithm of Le and Safey El Din (2020). We will explain the essential details of this subroutine later in Subsection 4.2.…”

Section: Description Of the Algorithmmentioning

confidence: 99%

“…The algorithm in Le and Safey El Din (2020) is based on constructing a symmetric matrix H F with entries in Q(y) associated to F . This matrix is basically a parametric version of the classical Hermite matrix for the ideal F (see, e.g., (Basu et al, 2006, Chap.…”

Section: Real Root Classificationmentioning

confidence: 99%

“…This allows to reduce our problem to zero-dimensional polynomial systems in Q(y) [x]. Next, we call to the algorithm given in the paper of Le and Safey El Din (2020) to compute semi-algebraic formulas that describe the projections of these systems on the y-space. Taking the union of these formulas, we obtain a semialgebraic formula that describes a dense subset of the interior of π(V R ).…”

mentioning

confidence: 99%

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Faster One Block Quantifier Elimination for Regular Polynomial Systems of Equations

Le,

Din

2021

Preprint

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Quantifier elimination over the reals is a central problem in computational real algebraic geometry, polynomial system solving and symbolic computation. Given a semialgebraic formula (whose atoms are polynomial constraints) with quantifiers on some variables, it consists in computing a logically equivalent formula involving only unquantified variables. When there is no alternate of quantifier, one has a one block quantifier elimination problem.We design a new practically efficient algorithm for solving one block quantifier elimination problems when the input semi-algebraic formula is a system of polynomial equations satisfying some mild assumptions such as transversality. When the input is generic, involves s polynomials of degree bounded by D with n quantified variables and t unquantified ones, we prove that this algorithm outputs semi-algebraic formulas of degree bounded by D using O n 8 t D 3t+2 t+D t arithmetic operations in the ground field where D := n D s (D − 1) n−s+1 n s . In practice, it allows us to solve quantifier elimination problems which are out of reach of the state-of-the-art (up to 8 variables).

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Machine learning the real discriminant locus

Bernal¹,

Hauenstein

Mehta

et al. 2023

Journal of Symbolic Computation

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Solving parametric systems of polynomial equations over the reals through Hermite matrices

Cited by 8 publications

References 43 publications

Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry

Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry

Faster One Block Quantifier Elimination for Regular Polynomial Systems of Equations

Machine learning the real discriminant locus

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