Homotopy techniques for solving sparse column support determinantal polynomial systems

Labahn, George; Din, Mohab Safey El; Schost, Éric; Vu, Thi Xuan

doi:10.1016/j.jco.2021.101557

Cited by 10 publications

(15 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Rather than using classical methods for solving these polynomial systems, we use the symbolic homotopy method for weighted domains given in [31], as this algorithm is the best suited to handle a weighted-degree structure exhibited by such systems. Indeed, the polynomial ring arising from an orbit parameter λ, K[e 1 , .…”

Section: Cost Of the Critical Points Per Orbit Algorithmmentioning

confidence: 99%

“…, f τ ) be a sequence of polynomials in K[Y ] and G = [g i,j ] ∈ K[Y ] p×q a matrix of polynomials such that p ≤ q and m = q − p + τ + 1, and let V p (G, f ) denote the set of points in K at which all polynomials in f and all p-minors of G vanish. In [31], a symbolic homotopy algorithm for weighted domains is presented which constructs a symbolic homotopy from a generic start system to the system defining V p (G, f ) and then uses this to efficiently determine the isolated points of V p (G, f ).…”

Section: Solving Weighted Determinantal Systemsmentioning

confidence: 99%

“…The main theorem of [31], in the special case of weighted polynomial rings, is given in terms of a number of parameters. Let (γ 1 , .…”

Section: Solving Weighted Determinantal Systemsmentioning

confidence: 99%

“…In our case we can make use of a recent sparse symbolic homotopy method given in [31] specifically designed to handle determinantal systems over weighted polynomial rings, that is, multivariate polynomial rings where each variable has a weighted degree, which is a positive integer. These domains arise naturally for our orbits: the domain arising from an orbit parameter λ has variables e i,k which are defined corresponding to elementary symmetric polynomials η i,k ; since η i,k has degree k, the variable e i,k will naturally be assigned weight k.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computing critical points for invariant algebraic systems

Faugère¹,

Labahn²,

Din³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

Let K be a field and φ, f = (f 1 , . . . , f s ) in K[x 1 , . . . , x n ] be multivariate polynomials (with s < n) invariant under the action of S n , the group of permutations of {1, . . . , n}. We consider the problem of computing the points at which f vanish and the Jacobian matrix associated to f , φ is rank deficient provided that this set is finite.We exploit the invariance properties of the input to split the solution space according to the orbits of S n . This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in d s , n+d d and n s+1 where d is the maximum degree of the input polynomials. When d, s are fixed, this is polynomial in n while when s is fixed and d n this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.

show abstract

Section: Cost Of the Critical Points Per Orbit Algorithmmentioning

confidence: 99%

Section: Solving Weighted Determinantal Systemsmentioning

confidence: 99%

“…The main theorem of [31], in the special case of weighted polynomial rings, is given in terms of a number of parameters. Let (γ 1 , .…”

Section: Solving Weighted Determinantal Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computing critical points for invariant algebraic systems

Faugère¹,

Labahn²,

Din³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Pioneering works in [23,25] give algorithms to compute a zerodimensional parametrization to present the set…”

Section: Introductionmentioning

confidence: 99%

Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials

Vu¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Let K be a field and K[ 1 , . . . , ] be a multivariate polynomial ring. Given a sequence of polynomials = ( 1 , . . . , ) and a polynomial in K[ 1 , . . . , ] with < , we consider the problem of computing the set ( , ) of points at which vanishes and the Jacobian matrix of , with respect to 1 , . . . , does not have full rank. This plays an essential role in many application areas.In this paper we focus on an important class of polynomials, the ones invariant under the action of the signed symmetric group . We introduce a notion called hyperoctahedral representation to represent -invariant sets. We study the invariance properties of the input to split ( , ) according to the orbits of and then we design an algorithm whose output being a hyperoctahedral representation of ( , ). The runtime of our algorithm is polynomial in the total number of points described by the output.

show abstract