Squarefree numbers in sumsets

We solve a problem of Filaseta by proving, that if N is sufficiently large, A ⊆ [N], |A| > N/9 and A + Adoes not contain any squarefree integer then all elements of A are congruent to 0 (mod 4) or 2 (mod 4). In order to show the main result we characterize the structure of all dense sets, whose elements sum to no squarefree number.     

Mixed discriminants

The mixed discriminant of $n$ Laurent polynomials in $n$ variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an $A$ -discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plücker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.     

Equal values of trinomials revisited

A necessary and sufficient condition is given for an equation ax ^m + bx ⁿ + c = dy ^p + ey ^q to have infinitely many rational solutions with a bounded denominator, under the assumption that m > n > 0, p > q > 0, ab ≠ 0 ≠ de and either m > p > 2, or m = p > 2 and n ≥ q. In a previous paper there was an additional assumption (m, n) = (p, q) = 1.     

On equal values of trinomials

Using Bilu–Tichy’s theorem and decomposition properties of trinomials an ineffective criterion is proved for two trinomials to have infinitely many equal values at rationals with a bounded denominator.     

Squarefree Vertex Cover Algebras

In this paper we introduce squarefree vertex cover algebras and exhibit a duality for them. We study the question when these algebras are standard graded and when these algebras coincide with the ordinary vertex cover algebras. It is shown that this is the case for simplicial complexes corresponding to principal Borel sets. Moreover, the generators of these algebras are explicitly described.     

Multiplicities of discriminants

We compute the multiplicity of the discriminant of a line bundle ￡ over a nonsingular varietyS at a given sectionX, in terms of the Chern classes of ￡ and of the cotangent bundle ofS, and the Segre classes of the jacobian scheme ofX inS. ForS a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a term that depends on the Milnor numbers of the singularities ofX _red. Also, under certain hypotheses, we provide formulas for the “higher discriminants” that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be “small”. Supported in part by the Max-Planck-Institut für Mathematik     

On reducible trinomials,III

It is shown that if a trinomial has a trinomial factor then under certain conditions the cofactor is irreducible.     

On equal values of trinomials

Using Bilu–Tichy’s theorem and decomposition properties of trinomials an ineffective criterion is proved for two trinomials to have infinitely many equal values at rationals with a bounded denominator.     

The Primitive Permutation Groups of Squarefree Degree

The paper gives lists of all the primitive permutation groupsof squarefree degree. All such groups are either solvable andact on a prime number of points, or are almost simple. Amongthe almost simple examples, the groups of Lie type have rankat most 2, or the point stabilizer is a parabolic subgroup.2000 Mathematics Subject Classification 20B15.     

Linearized trinomials with maximum kernel

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let q be a prime power, n be a positive integer and σ be a generator of $\mathrm{Gal}({\mathbb{F}}_{q^n}:{\mathbb{F}}_q)$. In this paper we provide closed formulas for the coefficients of a σ-trinomial f over ${\mathbb{F}}_{q^n}$ which ensure that the dimension of the kernel of f equals its σ-degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characterize those having σ-degree 3 and 4. Our techniques rely on the tools developed in [24]. Finally, we apply these results to investigate a class of rank metric codes introduced in [8], to construct quasi-subfield polynomials and cyclic subspace codes, obtaining new explicit constructions to the conjecture posed in [37].     

Permutation trinomials over F2m

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over ${\mathbb{F}}_{2^m}$ in Zieve's paper [30]. We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over ${\mathbb{F}}_{2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.     

Higher discriminants of binary forms

We propose a method for constructing systems of polynomial equations that define submanifolds of degenerate binary forms of an arbitrary degeneracy degree. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 147–157, November, 2007.     

Bounds of discriminants of number fields

In this survey, we introduce some progress on bounds of discriminants of number fields which contains the Rogers-Mulholland’s improvement of Minkowski’s classic inequality obtained by using geometry of numbers, Stark’s analytic method, improvement of analytic method by Odlyzko, and our results respectively.