Defective dual varieties for real spectra

Forsgård, Jens

doi:10.1007/s10801-018-0816-4

Cited by 7 publications

(8 citation statements)

References 12 publications

(23 reference statements)

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“…In more general situations, conditions for defectivity were given in [CC07], [DFS07], [Est10], [Ito15]. In particular, a complete characterization in terms of so-called iterated circuits was presented by Esterov [Est10] and proven in [Est18] (see also [For19] for a more general version). Recently, a different characterization was obtained by Furukawa and Ito [FI16] phrased in terms of so-called Cayley sums (we refer the reader to Section 2 for the definition of Cayley sums).…”

Section: Christopher Borger and Benjamin Nillmentioning

confidence: 99%

On defectivity of families of full-dimensional point configurations

Borger

Nill

2020

Proc. Amer. Math. Soc. Ser. B

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The mixed discriminant of a family of point configurations can be considered as a generalization of the A-discriminant of one Laurent polynomial to a family of Laurent polynomials. Generalizing the concept of defectivity, a family of point configurations is called defective if the mixed discriminant is trivial. Using a recent criterion by Furukawa and Ito we give a necessary condition for defectivity of a family in the case that all point configurations are full-dimensional. This implies the conjecture by Cattani, Cueto, Dickenstein, Di Rocco, and Sturmfels that a family of n full-dimensional configurations in Z n is defective if and only if the mixed volume of the convex hulls of its elements is 1.

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Section: Christopher Borger and Benjamin Nillmentioning

confidence: 99%

On defectivity of families of full-dimensional point configurations

Borger

Nill

2020

Proc. Amer. Math. Soc. Ser. B

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“…For the proof, see Corollary 3.23. Besides the relation to Galois theory, this result may be important as an illustration of a new approach to dual defectiveness in the toric setting, independent of the known ones [DiR06, DFS07, CC07, Est18, FI16, For17].…”

Section: Introductionmentioning

confidence: 95%

Galois theory for general systems of polynomial equations

Esterov¹

2019

Compositio Math.

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on the occasion of his 70th birthday. AbstractWe prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.In particular, our result proves the multivariate version of the Abel-Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive.The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

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“…The support set A is said to be dual defective if the variety XA is not a hypersurface. Dual defective support sets were characterized combinatorially by Esterov [Est10] in the integral case and by the first author [For18] in the real case. In the integral case, if A is nondefective, then the A-discriminant variety is defined by an integral polynomial D A P Zras, unique up to scaling, called the A-discriminant polynomial .…”

Section: The Positive Discriminantmentioning

confidence: 99%

“…The reader familiar with the theory of A-discriminants is aware that the dimension of the A-discriminantal variety might "unexpectedly" drop, in which case A is said to be dual-defective. The combinatorial characterization of defective support sets was completed by Esterov [Est10] in 2010 for the integral case and by the first author [For18] in 2018 in the real case. Similarly, the dimension of the Λ-discriminant D Λ might unexpectedly drop, and we leave it as an open combinatorial problem to characterize the dimension of D Λ in terms of the regular subdivision Λ.…”

Section: Introductionmentioning

confidence: 99%

The algebraic boundary of the sonc cone

Forsgård¹,

Wolff²

2019

Preprint

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We describe the algebraic boundary of the cone of sums of nonnegative circuit polynomials (sonc). This cone is generated by monomials and singular circuit polynomials. We interpret a singular circuit polynomial as compositions of an agiform in the sense of Reznick and a real torus action. Our main result is that the algebraic pieces of the boundary of the sonc cone are parametrized by families of tropical hypersurfaces. Each piece is contained in a rational variety called the positive discriminant, defined as the image of a Horn-Kapranov-type map. We give a complete description of the semi-algebraic stratification of the boundary of the sonc cone in the univariate case.We recover that the sonc cone equals the sage cone, and we prove that that the sonc cone is equal to the nonnegativity cone in codimension at most two. We also derive a combinatorial characterization of support sets for which the sonc cone is equal to the nonnegativity cone, disproving in the process a conjecture by Chandrasekaran, Murray, and Wierman.

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Defective dual varieties for real spectra

Cited by 7 publications

References 12 publications

On defectivity of families of full-dimensional point configurations

On defectivity of families of full-dimensional point configurations

Galois theory for general systems of polynomial equations

The algebraic boundary of the sonc cone

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