

Published

Online first articlesArticles are posted here individually soon after proof is returned from authors, before the corresponding journal issue is completed. All articles are in their final form, including issue number and pagination. Currently this page contains articles for: Vol. 64, no. 2 (2023)

Preface.  
Univariate
rational sums of squares.
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 nonnegative 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 nonnegative at the
real roots of a nonzero polynomial $f$ when the above assumption is
satisfied.

215–237 
A note on Bernstein–Sato ideals.
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.

239–246 
On
essential selfadjointness of singular
Sturm–Liouville operators.
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 comparisontype results in the spirit of Kurss to derive
criteria for $\tau_{\alpha}$ to be in the limitpoint and limitcircle case
at $x=0$. More precisely, if $\alpha \in \mathbb{R}$ and, for $0 < x$
sufficiently small,
\[
q(x) \geq [(3/4)(\alpha/2)]x^{\alpha2},
\]
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^{\alpha2}  (1/2) (2  \alpha) x^{\alpha2}
\sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{1}
\\
& \quad\quad\quad +[(3/4)+\varepsilon] x^{\alpha2}[\ln_{1}(x)]^{2},
\end{align*}
then $\tau_{\alpha}$ is nonoscillatory and in the limitpoint 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 limitcircle
case at $x=0$.
We also discuss a multidimensional 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 $R \in (0, \infty)$.

247–269 
Inequivalent representations of the dual space.
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.

271–280 
Normal form
transformations for modulated deepwater gravity waves.
Modulation theory is a wellknown tool to describe the longtime evolution
and stability of smallamplitude, 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
threedimensional deepwater surface waves, this approach leads to a Dysthe
equation that preserves the Hamiltonian character of the water wave
problem. An explicit calculation of the thirdorder Birkhoff normal form
that eliminates all nonresonant cubic terms yields a nonperturbative
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.

281–308 
Positivities in
Hall–Littlewood expansions and related plethystic operators.
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.

309–331 
Homogeneous weight enumerators over integer residue rings and failures of the MacWilliams identities.
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.

333–353 
Invariants
of formal pseudodifferential operator algebras and
algebraic modular forms.
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.

1–31 
On the zeros of univariate Epolynomials.
We consider two problems concerning real zeros of univariate Epolynomials.
First, we prove an explicit upper bound for the absolute values of the
zeroes of an Epolynomial 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 Epolynomials.
This result gives, in particular, an upper bound for the number of real
zeroes of an Epolynomial. We show this bound is sharp for particular
families of these functions, which proves that a conjecture by D. Richardson is
false.

33–46 
A generalization of the annihilating ideal graph for modules.
We show that an $R$module $M$ is noetherian (resp., artinian) if and only if
its annihilating submodule graph, $\mathbb{G}(M)$, is a nonempty 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.

47–65 
Published by the Unión Matemática Argentina
ISSN 16699637 (online), ISSN 00416932 (print)
Registro Nacional de la Propiedad Intelectual nro. 180.863
Contact: revuma(at)criba.edu.ar