The membrane inclusions curvature equations

Faugère, Jean-Charles; Hering, Milena; Phan, Jeff

doi:10.1016/s0196-8858(03)00039-3

Cited by 6 publications

(6 citation statements)

References 9 publications

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“…To give an estimation of the complexities of Abelian-F 5 and Abelian-FGLM algorithms, we are interested in two quantities. Definition 6 We define the density of R G d in R d and the density of R G in R by…”

Section: General Facts About the Ring Of Invariantsmentioning

confidence: 99%

“…However, it remains an open issue to solve efficiently the system in the general case. In the biology problem [6] or in the physics problem [9], an approach has been proposed if the group G is the symmetric group or copies of the symmetric group (elements of the form (σ ,...,σ ) ∈ S j m with m j = n.) MAIN RESULTS. We present efficient algorithms together with complexity analysis to solve polynomial systems which are globally invariant under the action of any commutative group G. The algorithms are based on three main ideas: first, since the group G is commutative, it is possible to diagonalize the group G, assuming that the characteristic of the field k and |G| are coprime.…”

Section: Introductionmentioning

confidence: 99%

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Gröbner bases of ideals invariant under a commutative group

Faugère

Svartz

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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We propose efficient algorithms to compute the Gröbner basis of an ideal I ⊂ k[x 1 ,...,x n ] globally invariant under the action of a commutative matrix group G, in the non-modular case (where char(k) doesn't divide |G|). The idea is to simultaneously diagonalize the matrices in G, and apply a linear change of variables on I corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on I are diagonal. This action induces a grading on the ring R = k[x 1 ,...,x n ], compatible with the degree, indexed by a group related to G, that we call G-degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into |G| submatrices of roughly the same size. In the same way, we are able to split the canonical basis of R/I (the staircase) if I is a zero-dimensional ideal. Therefore, we derive abelian versions of the classical algorithms F 4 , F 5 or FGLM. Moreover, this new variant of F 4 /F 5 allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of F 4 . Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time.

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Section: General Facts About the Ring Of Invariantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gröbner bases of ideals invariant under a commutative group

Faugère

Svartz

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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“…However it must be emphasized that polynomial systems arising in applications are very often highly structured. For instance, in several algebraic problems coming from applications the solutions (the algebraic variety) is invariant under the action of a finite group: for example the Cyclic-n problem [10], in Cryptography the NTRU Cryptosystem [12] or the membrane inclusions curvature equations in biology [6]. Hence an important subproblem is to solve efficiently such algebraic problems.…”

Section: Introductionmentioning

confidence: 99%

“…This method is very efficient if the Hironaka Decomposition of the ring of invariants is simple, but for the Cyclic-n problem, for example, it seems better to use a second method based on SAGBI Gröbner Basis techniques [8]. The second class of problems, which is probably the most important in practice, is to consider polynomial systems of which the set of solutions is globally invariant under the action of a finite group (this is the case of the biology example [6]). The goal of the present paper is to propose an efficient method to solve such problems assuming that the group is the whole symmetric group.…”

Section: Introductionmentioning

confidence: 99%

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Solving polynomial systems globally invariant under an action of the symmetric group and application to the equilibria of N vortices in the plane

Faugère¹,

Svartz²

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

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We propose an efficient algorithm to solve polynomial systems of which equations are globally invariant under an action of the symmetric group S N acting on the variable x i with σ (x i) = x σ (i) and the number of variables is a multiple of N. For instance, we can assume that swapping two variables (or two pairs of variables) in one equation gives rise to another equation of the system (perhaps changing the sign). The idea is to apply many times divided difference operators to the original system in order to obtain a new system of equations involving only the symmetric functions of a subset of the variables. The next step is to solve the system using Gröbner techniques; this is usually several order faster than computing the Gröbner basis of the original system since the number of solutions of the corresponding ideal, which is always finite has been divided by at least N!. To illustrate the algorithm and to demonstrate its efficiency, we apply the method to a well known physical problem called equilibria positions of vortices. This problem has been studied for almost 150 years and goes back to works by von Helmholtz and Lord Kelvin. Assuming that all vortices have same vorticity, the problem can be reformulated as a system of polynomial equations invariant under an action of S N. Using numerical methods, physicists have been able to compute solutions up to N ≤ 7 but it was an open challenge to check whether the set of solution is complete. Direct naive approach of Gröbner bases techniques give rise to hard-to-solve polynomial system: for instance, when N = 5, it takes several days to compute the Gröbner basis and the number of solutions is 2060. By contrast, applying the new algorithm to the same problem gives rise to a system of 17 solutions that can be solved in less than 0.1 sec. Moreover, we are able to compute all equilibria when N ≤ 7.

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Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

Anthony

Grant

Gritzmann

et al. 2014

Mathematics and Visualization

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We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Dedicated to Tien-Yien Li, in honor of his birthday.

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The membrane inclusions curvature equations

Cited by 6 publications

References 9 publications

Gröbner bases of ideals invariant under a commutative group

Gröbner bases of ideals invariant under a commutative group

Solving polynomial systems globally invariant under an action of the symmetric group and application to the equilibria of N vortices in the plane

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

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