Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)
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Minimal prime ideals of skew Hurwitz series ring
Full text linkLet R be a ring with an endomorphism α. In this paper we obtain necessary and sufficient conditions on R and α such that the skew Hurwitz series ring (HR, α) is a 2-primal ring. In particular, it is proved that, under suitable conditions, (HR,α) is 2-primal if and only if for every minimal prime ideal P∗ in (HR, α) there exists a minimal prime ideal P of R such that P is completely prime and P∗=(HP,α) if and only if P((HR,α))=(H(nil(R)),α) if and only if R is 2-primal and nil((HR,α))=(H(nil(R)),α) if and only if every minimal α-prime ideal of R is completely prime
Sequential approximate optimality conditions for multi objective fractional programming problems via sequential approximate subdifferential calculus
No full textThe purpose of this paper is to establish sequential efficient optimality conditions, without any constraint qualification, characterizing an efficient solution for multiobjec- tive fractional programming problem. The approach used in this investigation is based on sequential subdifferential calculus. By using the same approach, we establish the standard optimality conditions under a constraint qualification. Finally, we present an example illustrating the main result of this paper
Reciprocal maximum likelihood degrees of diagonal linear concentration models
Full text linkWe show that the reciprocal maximal likelihood degree (rmld) of a diagonal linear concentration model L ⊆ Cn of dimension r is equal to (-2)rχM(1/2), where χM is the characteristic polynomial of the matroid M associated to L. In particular, this establishes the polynomiality of the rmld for general diagonal linear concentration models, positively answering a question of Sturmfels, Timme, and Zwiernik
Duplicate, Bernstein algebras and evolution algebras
Full text linkIn this paper, we firstly study a commutative algebra E over a field F of Char(F) ̸= 2 that satisfying dim(E2) = 1. We show that, such an algebra is an evolution algebra. Afterwards, we pay attention to commutative duplicate of a commutative algebra E. We find necessary and sufficient condition in which the duplicate D(E) is an evolution algebra. And, we finish by studying an evolution algebra that is a Bernstein algebra. We classify that algebras, up to isomorphism, in dimension ≤ 4.
{\bf Keywords}: Evolution algebras, Bernstein algebras, Duplicate, natural base
Cubic surfaces on the singular locus of the Eckardt hypersurface
Full text linkThe Eckardt hypersurface in P19 is the closure of the locus of smooth cubic surfaces with an Eckardt point, which is a point common to three of the 27 lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular locus of the model of this hypersurface in P4, obtained via restriction to the space of cubic surfaces possessing a so-called Sylvester form. We prove that, inside the moduli of cubics, the singular locus corresponds to a reducible surface with two rational irreducible components intersecting along two rational curves. The two curves intersect at two points representing the Clebsch and the Fermat cubic surfaces. We observe that the cubic surfaces parameterized by the two components or the two rational curves are distinguished by the number of Eckardt points and automorphism groups
Towards classifying toric degenerations of cubic surfaces
Full text linkWe investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and who leave the question of classifying these degenerations in the degree 3 case as an open problem. In order to carry out this classification we describe an approach which is closely related to tropical geometry and present partial results in this direction.
 
Determinantal representations of the cubic discriminant
Full text linkWe compute and study two determinantal representations of the discriminant of a cubic quaternary form. The first representation is the Chow form of the 2-uple embedding of P3 and is computed as the Pfaffian of the Chow form of a rank 2 Ulrich bundle on this Veronese variety. We then consider the determinantal representation described by Nanson. Weinvestigate the geometric nature of cubic surfaces whose discriminant matrices satisfy certain rank conditions. As a special case of interest, we use certain minors of this matrix to suggest equations vanishing on the locus of -nodal cubic surfaces
The Newton polytope of the discriminant of a quaternary cubic form
Full text linkWe determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with D-equivalence classes of regular triangulations of the 3-dilated tetrahedron. We describe how to compute these triangulations and their D-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer number of triangulations effectively, as well as devising a suitably fast algorithm for computation of a D-equivalence class
An octanomial model for cubic surfaces
Full text linkWe present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms, written in moduli from the E6 hyperplane arrangement. If such a surface is tropically smooth then its 27 tropical lines are distinct. We focus on explicit computations, both symbolic and p-adic numerical
On the eigenpoints of cubic surfaces
Full text linkWe show that the eigenschemes of 4 × 4 × 4 symmetric tensors are parameterized by a linear subvariety of the Grassmannian Gr(3, P14). We also study the decomposition of the eigenscheme into the subscheme associated to the zero eigenvalue and its residue. In particular, we describe the possible degrees and dimensions