New and old results in resultant theory

Морозов, А.; Shakirov, Sh.

doi:10.1007/s11232-010-0044-0

Cited by 42 publications

(36 citation statements)

References 20 publications

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“…This question was addressed in [100] in association with the work by A. Connes and D. Kreimer [101][102][103][104][105][106][107] (which describes the Bogoliubov-Zimmermann formulas in terms of the Hopf algebra of Feynman diagrams). The true motivation was, however, somewhat broader and included also the search for the QFT reformulation of the problems of non-linear algebra [108][109][110][111]. In the present paper, we discuss further steps towards constructing the renormalization group (RG) complete models and the RG-closed sets of JHEP06 (2017)115 operators.…”

Section: Jhep06(2017)115mentioning

confidence: 99%

Ward identities and combinatorics of rainbow tensor models

Itoyama¹,

Миронов²,

Морозов³

2017

J. High Energ. Phys.

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Abstract:We discuss the notion of renormalization group (RG) completion of nonGaussian Lagrangians and its treatment within the framework of Bogoliubov-Zimmermann theory in application to the matrix and tensor models. With the example of the simplest non-trivial RGB tensor theory (Aristotelian rainbow), we introduce a few methods, which allow one to connect calculations in the tensor models to those in the matrix models. As a byproduct, we obtain some new factorization formulas and sum rules for the Gaussian correlators in the Hermitian and complex matrix theories, square and rectangular. These sum rules describe correlators as solutions to finite linear systems, which are much simpler than the bilinear Hirota equations and the infinite Virasoro recursion. Search for such relations can be a way to solving the tensor models, where an explicit integrability is still obscure.

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Section: Jhep06(2017)115mentioning

confidence: 99%

Ward identities and combinatorics of rainbow tensor models

Itoyama¹,

Миронов²,

Морозов³

2017

J. High Energ. Phys.

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“…Formula (1) relating the Lebesgue volume G 1 with exp(−g) is already proved (with a different argument) in Morozov and Shakirov [18], [19] where the goal of the authors is to try to express the non Gaussian integral exp(−g) in terms of algebraic invariants of g.…”

Section: B Some Preliminary Resultsmentioning

confidence: 98%

A generalization of the Löwner-John's ellipsoid theorem

Lasserre

2013

52nd IEEE Conference on Decision and Control

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We provide the following generalization of the Löwner-John's ellipsoid theorem. Given a (non necessarily convex) compact set K ⊂ R n and an even integer d ∈ N, there is a unique homogeneous polynomial g of degree d such that K ⊂ G := {x : g(x) ≤ 1} and G has minimum volume among all such sets. The symmetric case of the Lowner-John theorem is a particular case when d = 2, and importantly, we neither require the set K nor the sublevel set G to be convex. We also provide a numerical scheme to approximate the optimal value and the unique optimal solution as closely as desired.52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy 978-1-4673-5717-3/13/$31.00 ©2013 IEEE

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“…For the sparse resultant, a bevy of results have established matrix and, more particularly, determinantal formulae for a large class of systems. An introductory survey can be found in [28].…”

Section: Resultant Formulaementioning

confidence: 99%

Compact Formulae in Sparse Elimination

Emiris

2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination.We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bézout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.

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New and old results in resultant theory

Cited by 42 publications

References 20 publications

Ward identities and combinatorics of rainbow tensor models

Ward identities and combinatorics of rainbow tensor models

A generalization of the Löwner-John's ellipsoid theorem

Compact Formulae in Sparse Elimination

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New and old results in resultant theory

Cited by 42 publications

References 20 publications

Ward identities and combinatorics of rainbow tensor models

Ward identities and combinatorics of rainbow tensor models

A generalization of the L&#x00F6;wner-John's ellipsoid theorem

Compact Formulae in Sparse Elimination

Contact Info

Product

Resources

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A generalization of the Löwner-John's ellipsoid theorem