Rendiconti del Seminario Matematico
della Università di Padova
The Mathematical Journal of the University of Padua
e-ISSN: 2240-2926
Forthcoming Articles
(Last update: November 12, 2019)The following articles will soon be published in this journal. Their publication status can be :
- online first – final version available;
- accepted – manuscript available.
Click on the title to view the paper.
Manuscripts of the accepted Papers
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Zero divisors of support size 3 in group algebras and trinomials divided by irreducible polynomials over GF(2)
Abstract: A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length of possible ones, where by the length we mean the size of the support of an element of the group algebra. The case length $2$ cannot be happen. The first unsettled case is the existence of zero divisors of length $3$. Here we study possible length $3$ zero divisors in the rational group algebras and in the group algebras over the field $\mathbb{F}_p$ with $p$ elements for some prime $p$. As a consequence we prove that the rational group algebras of torsion-free groups which are residually finite $p$-group for some prime $p\neq 3$ have no zero divisor of length $3$. We note that the determination of all zero divisors of length $3$ in group algebras over $\mathbb{F}_2$ of cyclic groups is equivalent to find all trinomials (polynomials with 3 non-zero terms) divided by irreducible polynomials over $\mathbb{F}_2$. The latter is a subject studied in coding theory and we add here some results, e.g. we show that $1+x+x^2$ is a zero divisor in the group algebra over $\mathbb{F}_2$ for some element $x$ of the group if and only if $x$ is of finite order divided by $3$ and we find all $\beta$ in the group algebra of the shortest length such that $(1+x+x^2)\beta=0$; and $1+x^2+x^3$ or $1+x+x^3$ is a zero divisor in the group algebra over $\mathbb{F}_2$ for some element $x$ of the group if and only if $x$ is of finite order divided by $7$.
received 09.12.2018, accepted 04.11.2019 (14 pages)
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On the F -norm of a finite group
Abstract: Let $G$ be a finite group and ${\mathcal{F}}$ be a non-empty formation. We define the ${\mathcal{F}^*}$-norm, denoted by $N_{\mathcal{F}}^{*}(G)$, to be intersection of the normalizers of the ${\mathcal{F}}$-residuals of all $F$-subgroups of $G$, where $F={\mathcal{N}}{\mathcal{F}}$ is the class of all groups whose ${\mathcal{F}}$-residuals are nilpotent. In this paper, we research the properties of $N_{\mathcal{F}}^{*}(G)$ and investigate the relationship between $N_{\mathcal{F}}^{*}(G)$ and $N_{\mathcal{F}}(G),$ where $N_{\mathcal{F}}(G)$ is the intersection of the normalizers of the ${\mathcal{F}}$-residuals of all subgroups of $G.$ We show that $N_{\mathcal{F}}^{*}(G)=N_{\mathcal{F}}(G)$ if ${\mathcal{A}}\subseteq {\mathcal{F}}\subseteq{\mathcal{N}}.$
received 21.06.2018, accepted 28.10.2019 (10 pages)
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A natural fibration for rings
Abstract: A ringed partially ordered set with zero is a pair $(L, F)$, where $L$ is a partially ordered set with a least element $0_L$ and $F\colon L\to\mathbf{Ring}$ is a covariant functor. Here the partially ordered set $L$ is given a category structure in the usual way and $\mathbf{Ring}$ denotes the category of associative rings with identity. Let $\mathbf{RingedParOrd}_0$ be the category of ringed partially ordered sets with zero. There is a functor $\mathcal{H}\colon\mathbf{Ring}\to\mathbf{RingedParOrd}_0$ that associates to any ring $R$ a ringed partially ordered set with zero $(Hom(R), F_R)$. The functor $\mathcal{H}$ has a left inverse $Z\colon\mathbf{RingedParOrd}_0\to\mathbf{Ring}$. The category $\mathbf{RingedParOrd}_0$ is a fibred category.
received 27.02.2019, accepted 28.10.2019 (10 pages)
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Normalizers of classical groups arising under extension of the base ring
Abstract: Let $R$ be a unital subring of a commutative ring $S,$ which is a free $R$-module of rank $m.$ In 1994 and then in 2017, V. A. Koibaev and we described normalizers of subgroups $GL(n, S)$ and $E(n, S)$ in $G = GL(mn, R)$, and showed that they are equal and coincide with the set $\{g \in G: E(n, S)^g \leq GL(n, S)\} = Aut(S/R) \ltimes GL(n, S).$\mathcal{H}reover, for any proper ideal A of R, $$N_{G}(E(n, S)E(mn, R, A)) = \rho_{A}^{-1}(N_{GL(mn, R/A)}(E(n, S/SA))).$$ In the present paper, we prove similar results about normalizers of classical subgroups, namely, the normalizers of subgroups $EO(n, S), SO(n, S), O(n, S)$ and $GO(n,S)$ in $G$ are equal and coincide with the set $\{g \in G: EO(n, S)^g \leq GO(n, S)\} = Aut(S/R) \ltimes GO(n, S).$ Similarly, the ones of subgroups $Ep(n, S),$\\ $Sp(n, S)$ and $GSp(n, S)$ are equal and coincide with the set $\{g \in G: Ep(n, S)^g \leq GSp(n, S)\} = Aut(S/R) \ltimes GSp(n, S).$ Moreover, for any proper ideal $A$ of $R,$ $$N_{G}(EO(n, S)E(mn, R, A)) = \rho_{A}^{-1}(N_{GL(mn, R/A)}(EO(n, S/SA)))$$ and $$N_{G}(Ep(n, S)E(mn, R, A)) = \rho_{A}^{-1}(N_{GL(mn, R/A)}(Ep(n, S/SA))).$$ When $R = S,$ we obtain the known results of N. A. Vavilov and V. A. Petrov.
received 18.10.2018, accepted 10.10.2019 (13 pages)
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Hasse-Witt matrices for polynomials, and applications
Abstract: In a classical paper, Manin gives a congruence [15, Theorem 1] for the characteristic polynomial of the action of Frobenius on the Jacobian of a curve $C$, defined over the finite field $\mathbf{F}_{q}$, $q=p^m$, in terms of its Hasse-Witt matrix. The aim of this article is to prove a congruence similar to Manin's one, valid for any $L$-function $L(f,T)$ associated to the exponential sums over affine space attached to an additive character of $\mathbf{F}_q$, and a polynomial $f$. In order to do this, we define a Hasse-Witt matrix $\mathrm{HW}(f)$, which depends on the characteristic $p$, the set $D$ of exponents of $f$, and its coefficients. We also give some applications to the study of the Newton polygon of Artin-Schreier (hyperelliptic when $p=2$) curves, and zeta functions of varieties.
received 21.03.2019, accepted 04.10.2019 (21 pages)
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Some elementary questions in the calculus of variations
Abstract: We discuss some elementary questions in the calculus of variations related to the Lavrentiev phenomenon and the uniqueness of the minimizers.
received 17.07.2019, accepted 27.09.2019 (6 pages)
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Erdelyi-Kober fractional integral operators on ball Banach function spaces
Abstract: We establish the boundedness of the Erd\'{e}lyi-Kober fractional integral operators on ball Banach function spaces. In particular, it gives the boundedness of the Erd\'{e}lyi-Kober fractional integral operators on amalgam spaces and Morrey spaces.
received 23.07.2019, accepted 10.09.2019 (14 pages)
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A Monodromy Criterion for the Good Reduction of K3 Surfaces
Abstract: We give a criterion for the good reduction of semistable $K3$ surfaces over $p$-adic fields. We use neither $p$-adic Hodge theory nor transcendental methods as in the analogous proofs of criteria for good reduction of curves or $K3$ surfaces. We achieve our goal by realizing the special fiber $X_s$ of a semistable model $X$ of a $K3$ surface over the $p$-adic field $K$, as a special fiber of a log-family in characteristic $p$ and use an arithmetic version of the Clemens-Schmid exact sequence in order to obtain a Kulikov-Persson-Pinkham classification theorem in characteristic $p$.
received 23.04.2019, accepted 27.08.2019 (20 pages)
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Integrable derivations in the sense of Hasse-Schmidt for some binomial plane curves
Abstract: We describe the module of integrable derivations in the sense of Hasse-Schmidt of the quotient of the polynomial ring in two variables over an ideal generated by the equation $x^n-y^q$.
received 08.01.2019, accepted 08.07.2019 (12 pages)
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Composites in semirings of boolean groups
Abstract: We estimate the number of composite elements in the $n$-th grade of a group semiring of finite boolean groups. In view of this result we conjecture that the composites in these semirings of finite groups are thinly dispersed.
received 15.08.2018, accepted 02.07.2019 (6 pages)
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The flatness of ternary cyclotomic polynomials
Abstract: It is well known that all of the prime cyclotomic polynomials and binary cyclotomic polynomials are flat, and the flatness of ternary cyclotomic polynomials is much more complicated. Let $p
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Milnor's alternative for finitely generated relatively free groups that are locally graded
Abstract: Let $G$ be a finitely generated relatively free group that is locally graded. We show that either $G$ contains a non-trivial free subsemigroup or $G$ is nilpotent-by-finite.
received 30.05.2018, accepted 17.06.2019 (5 pages)
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On $\alpha$-minimizing hypercones
Abstract: In this paper we considerably extend the class of known $\alpha$-minimizing hypercones using sub-calibration methods. Indeed, the improvement of previous results follows from a careful analysis of special cubic and quartic polynomials.
received 29.10.2018, accepted 31.05.2019 (20 pages)
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Closed ideals in the uniform topology on the ring of real-valued continuous functions on a frame
Abstract: For a completely regular frame $L$, the ring $\mathcal{R}L$ of real-valued continuous functions on $L$ is equipped with the uniform topology. The closed ideals of $\mathcal{R}L$ in this topology are studied, and a new, merely algebraic characterization of these ideals is given. This result is used to describe the real ideals of $\mathcal{R}L$, and to characterize pseudocompact frames and Lindelöf frames. It is shown that a frame $L$ is finite if and only if every ideal of $\mathcal{R}L$ is closed. Finally, we prove that every closed ideal in $\mathcal{R}L$ is an intersection of maximal ideals.
received 31.01.2018, accepted 30.04.2019 (18 pages)
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Pseudo-dualizing complexes and pseudo-derived categories
Abstract: The definition of a pseudo-dualizing complex is obtained from that of a dualizing complex by dropping the injective dimension condition, while retaining the finite generatedness and homothety isomorphism conditions. In the specific setting of a pair of associative rings, we show that the datum of a pseudo-dualizing complex induces a triangulated equivalence between a pseudo-coderived category and a pseudo-contraderived category. The latter terms mean triangulated categories standing in between the conventional derived category and the coderived or the contraderived category. The constructions of these triangulated categories use appropriate versions of the Auslander and Bass classes of modules. The constructions of derived functors providing the triangulated equivalence are based on a generalization of a technique developed in our previous paper [45].
received 24.09.2018, accepted 30.04.2019 (77 pages)
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Sign-changing solutions of boundary value problems for semilinear $\Delta_\gamma$ Laplace equations
Abstract: In this article, we study the multiplicity of weak solutions to the boundary value problem \begin{eqnarray*} -G_\alpha u&=& g(x,y,u) + f(x,y,u) \ \mbox{ in }\ \ \Omega,\\ u&=&\ 0 \hskip 3cm \mbox{ on }\ \partial \Omega, \notag \end{eqnarray*} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2), \alpha \in \mathbb{N}, g(x,y,\xi),$ $ f(x,y,\xi) $ are Carathéodory functions and $ G_\alpha $ is the Grushin operator. We use the lower bounds of eigenvalues and an abstract theory on sign-changing solutions.
received 01.04.2019, accepted 19.04.2019 (16 pages)
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A notion analogous to the discriminant for transcendental elements in certain extensions of local elds
Abstract: Let $(K,|\cdot |)$ be a local field. In this paper we define an invariant analogous to the discriminant over $K$ for certain transcendental elements over $K$.
received 03.12.2018, accepted 28.03.2019 (8 pages)
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Divisibility and duo-rings
Abstract: This paper investigates the projective dimension of the maximal right ring of quotients $Q^r(R)$ of a right non-singular ring $R$. Our discussion addresses the question under which conditions $p.d.(Q) \leq 1$ guarantees that the module $Q/R$ is a direct sum of countably generated modules extending Matlis' Theorem for integral domains to a non-commutative setting.
received 11.12.2018, accepted 25.03.2019 (16 pages)
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