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Vol. 64, no. 2 (2023)
Special volume: Mathematical Congress of the Americas 2021

Editors: Guillermo Cortiñas (managing editor, Universidad de Buenos Aires); Renato Iturriaga (CIMAT, Guanajuato, Mexico); Andrea Nahmod (University of Massachusetts, Amherst, USA); Jimmy Petean (CIMAT, Guanajuato, Mexico); Arturo Pianzola (University of Alberta, Canada and Universidad CAECE, Argentina); Cecilia Salgado (University of Groningen, Netherlands and Universidade Federal do Rio de Janeiro, Brazil); Soledad Torres (CIMFAV-INGEMAT, Universidad de Valparaíso, Chile); Bernardo Uribe (Universidad del Norte, Colombia); Steven Weintraub (Lehigh University, Bethlehem, USA).

Preface. Guillermo Cortiñas
Univariate rational sums of squares. Teresa Krick, Bernard Mourrain, and Agnes Szanto
Given rational univariate polynomials $f$ and $g$ such that $\gcd (f,g)$ and $f/\gcd(f,g)$ are relatively prime, we show that $g$ is non-negative at all the real roots of $f$ if and only if $g$ is a sum of squares of rational polynomials modulo $f$. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial $g$ is non-negative at the real roots of a non-zero polynomial $f$ when the above assumption is satisfied.
A note on Bernstein–Sato ideals. Josep Àlvarez Montaner
We define the Bernstein–Sato ideal associated to a tuple of ideals and we relate it to the jumping points of the corresponding mixed multiplier ideals.
On essential self-adjointness of singular Sturm–Liouville operators. S. Blake Allan, Fritz Gesztesy, and Alexander Sakhnovich
Considering singular Sturm–Liouville differential expressions of the type \[ \tau_{\alpha} = -(d/dx)x^{\alpha}(d/dx) + q(x), \quad x \in (0,b), \, \alpha \in \mathbb{R}, \] we employ some Sturm comparison-type results in the spirit of Kurss to derive criteria for $\tau_{\alpha}$ to be in the limit-point and limit-circle case at $x=0$. More precisely, if $\alpha \in \mathbb{R}$ and, for $0 < x$ sufficiently small, \[ q(x) \geq [(3/4)-(\alpha/2)]x^{\alpha-2}, \] or, if $\alpha\in (-\infty,2)$ and there exist $N\in\mathbb{N}$ and $\varepsilon>0$ such that, for $0 < x$ sufficiently small, \begin{align*} & q(x)\geq[(3/4)-(\alpha/2)]x^{\alpha-2} - (1/2) (2 - \alpha) x^{\alpha-2} \sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{-1} \\ & \quad\quad\quad +[(3/4)+\varepsilon] x^{\alpha-2}[\ln_{1}(x)]^{-2}, \end{align*} then $\tau_{\alpha}$ is nonoscillatory and in the limit-point case at $x=0$. Here iterated logarithms for $0 < x$ sufficiently small are of the form \[ \ln_1(x) = |\ln(x)| = \ln(1/x), \qquad \ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in \mathbb{N}. \] Analogous results are derived for $\tau_{\alpha}$ to be in the limit-circle case at $x=0$. We also discuss a multi-dimensional application to partial differential expressions of the type \[ - \operatorname{div} |x|^{\alpha} \nabla + q(|x|), \quad \alpha \in \mathbb{R}, \, x \in B_n(0;R) \backslash \{0\}, \] with $B_n(0;R)$ the open ball in $\mathbb{R}^n$, $n\in \mathbb{N}$, $n \geq 2$, centered at $x=0$ of radius $R \in (0, \infty)$.
Inequivalent representations of the dual space. Tepper L. Gill, Douglas Mupasiri, and Erdal Gül
We show that there exist inequivalent representations of the dual space of $\mathbb{C}[0,1]$ and of $L_p[\mathbb{R}^n]$ for $p \in [1,\infty)$. We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if $\mathcal{A}$ is a proper closed subspace of $\mathbb{C}[0,1]$, there always exists a closed subspace $\mathcal{A}^\bot$ (with the same definition as for $L_2[0,1]$) such that $\mathcal{A} \cap \mathcal{A}^\bot = \{0\}$ and $\mathcal{A} \oplus \mathcal{A}^\bot = \mathbb{C}[0,1]$. Thus, the geometry of $\mathbb{C}[0,1]$ can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator $A$ on $\mathbb{C}[0,1]$ has a uniquely defined adjoint $A^*$ defined on $\mathbb{C}[0,1]$, and both can be extended to bounded linear operators on $L_2[0,1]$. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to $L_p[\mathbb{R}^n]$. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on $\mathbb{C}[0,1]$ is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.
Normal form transformations for modulated deep-water gravity waves. Philippe Guyenne, Adilbek Kairzhan, and Catherine Sulem
Modulation theory is a well-known tool to describe the long-time evolution and stability of small-amplitude, oscillating solutions to dispersive nonlinear partial differential equations. There have been a number of approaches to deriving envelope equations for weakly nonlinear waves. Here we review a systematic method based on Hamiltonian transformation theory and averaging Hamiltonians. In the context of the modulation of two- or three-dimensional deep-water surface waves, this approach leads to a Dysthe equation that preserves the Hamiltonian character of the water wave problem. An explicit calculation of the third-order Birkhoff normal form that eliminates all non-resonant cubic terms yields a non-perturbative procedure for the reconstruction of the free surface. We also present new numerical simulations of this weakly nonlinear approximation using the version with exact linear dispersion. We compare them against computations from the full water wave system and find very good agreement.
Positivities in Hall–Littlewood expansions and related plethystic operators. Marino Romero
The Hall–Littlewood polynomials $\mathbf{H}_\lambda = Q'_\lambda[X;q]$ are an important symmetric function basis that appears in many contexts. In this work, we give an accessible combinatorial formula for expanding the related symmetric functions $\mathbf{H}_\alpha$ for any composition $\alpha$, in terms of the complete homogeneous basis. We do this by analyzing Jing's operators, which extend to give nice expansions for the related symmetric functions $\mathbf{C}_\alpha$ and $\mathbf{B}_\alpha$ which appear in the formulation of the Compositional Shuffle Theorem. We end with some consequences related to eigenoperators of the modified Macdonald basis.
Homogeneous weight enumerators over integer residue rings and failures of the MacWilliams identities. Jay A. Wood
The MacWilliams identities for the homogeneous weight enumerator over $\mathbb{Z}/m\mathbb{Z}$ do not hold for composite $m \geq 6$. For such $m$, there exist two linear codes over $\mathbb{Z}/m\mathbb{Z}$ that have the same homogeneous weight enumerator, yet whose dual codes have different homogeneous weight enumerators.

Vol. 65, no. 1 (2023)

Invariants of formal pseudodifferential operator algebras and algebraic modular forms. François Dumas and François Martin
We study the question of extending an action of a group $\Gamma$ on a commutative domain $R$ to a formal pseudodifferential operator ring $B=R(\!(x\,;\,d)\!)$ with coefficients in $R$, as well as to some canonical quadratic extension $C=R(\!(x^{1/2}\,;\,\frac 12 d)\!)_2$ of $B$. We give conditions for such an extension to exist and describe under suitable assumptions the invariant subalgebras $B^\Gamma$ and $C^\Gamma$ as Laurent series rings with coefficients in $R^\Gamma$. We apply this general construction to the numbertheoretical context of a subgroup $\Gamma$ of $\mathrm{SL}(2,\mathbb{C})$ acting by homographies on an algebra $R$ of functions in one complex variable. The subalgebra $C_0^\Gamma$ of invariant operators of nonnegative order in $C^\Gamma$ is then linearly isomorphic to the product space $\mathcal{M}_0=\prod_{j\geq 0}M_j$, where $M_j$ is the vector space of algebraic modular forms of weight $j$ in $R$. We obtain a structure of noncommutative algebra on $\mathcal{M}_0$, which can be identified with a space of algebraic Jacobi forms. We study properties of the correspondence $\mathcal{M}_0\to C_0^\Gamma$, whose restriction to even weights was previously known, using arithmetical arguments and the algebraic results of the first part of the article.
On the zeros of univariate E-polynomials. María Laura Barbagallo, Gabriela Jeronimo, and Juan Sabia
We consider two problems concerning real zeros of univariate E-polynomials. First, we prove an explicit upper bound for the absolute values of the zeroes of an E-polynomial defined by polynomials with integer coefficients that improves the bounds known up to now. On the other hand, we extend the classical Budan–Fourier theorem for real polynomials to E-polynomials. This result gives, in particular, an upper bound for the number of real zeroes of an E-polynomial. We show this bound is sharp for particular families of these functions, which proves that a conjecture by D. Richardson is false.
A generalization of the annihilating ideal graph for modules. Soraya Barzegar, Saeed Safaeeyan, and Ehsan Momtahan
We show that an $R$-module $M$ is noetherian (resp., artinian) if and only if its annihilating submodule graph, $\mathbb{G}(M)$, is a non-empty graph and it has ascending chain condition (resp., descending chain condition) on vertices. Moreover, we show that if $\mathbb{G}(M)$ is a locally finite graph, then $M$ is a module of finite length with finitely many maximal submodules. We also derive necessary and sufficient conditions for the annihilating submodule graph of a reduced module to be bipartite (resp., complete bipartite). Finally, we present an algorithm for deriving both $\Gamma (\mathbb{Z}_n)$ and $\mathbb{G}(\mathbb{Z}_n)$ by Maple, simultaneously.