Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.02068

The local-to-global minimum problem of Calvert and Vamanamurthy: a quartic counterexample in three variables and the cubic case in all dimensions

Hugo Aduén
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1980 Calvert and Vamanamurthy asked whether a polynomial whose only critical point is a local minimum must attain its global minimum there. They gave counterexamples of every degree at least five in two or more variables and settled cubics in two and three variables and quartics in two variables, leaving open quartics in three or more variables and cubics in four or more variables. We resolve both remaining cases. First, we give an explicit quartic in three variables whose unique critical point is a nondegenerate strict local minimum, although the polynomial is unbounded below; the example extends to every dimension at least three, and its critical locus is a single point even over the complex numbers. Second, we prove that in every dimension a polynomial of exact degree three with a local minimum has another critical point, so that the cubic case holds vacuously; the proof is elementary and avoids the topological degree arguments used previously in two and three variables. Together with the results of Calvert and Vamanamurthy, this gives the complete classification of the local-to-global property by degree and dimension.

Explore related subjects

math.CAmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hugo Aduén. 2026-11-01. The local-to-global minimum problem of Calvert and Vamanamurthy: a quartic counterexample in three variables and the cubic case in all dimensions. https://arxiv.org/abs/2611.02068

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Attainable forms of lower spectra

Let $d\in\mathbb{N}$ and $φ\colon(0,1)\to[0,d]$. We prove there exists a set $F\subset\mathbb{R}^d$ whose lower spectrum $\operatorname{dim}^θ_{\mathrm{L}} F$ satisfies $(1-θ)\operatorname{dim}^θ_{\mathrm{L}} F = φ(θ)$ for all $θ\in(0,1)$ if and only if for all $λ,θ\in(0,1)$, \begin{equation*} φ(θ) \leq φ(λθ) - θφ(λ) \leq (1-θ) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^θ_{\mathrm{L}} F$. In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgoványi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions.

math.CA↗
arXiv2026

A $Tb$ type theorem for suppressed kernels

In this article, a non-homogeneous $Tb$ type theorem for arbitrary dimensional Calderón-Zygmund singular integral operators is proved. This is an extension of an analogous non-homogeneous $Tb$ theorem for the Cauchy transform, in the planar setting, due to Nazarov, Treil and Volberg. The novelties of the present work are the change of dimension and the fact that the operators to which the theorem applies are not necessarily antisymmetric. The techniques used in the proof include, among others, suppressed kernels, decompositions in $L^2(μ)$, where $μ$ is a Radon measure in $\mathbb{R}^d$, and a probabilistic argument resulting from taking averages of the operators involved.

math.CA↗
arXiv2026

Sparse domination modulo polynomials

We develop a sparse domination theory that captures both local polynomial approximation and cancellation of the input below the scale of each sparse cube. Its coefficients are percentiles of localized smooth maximal functions of the input minus its local polynomial approximation, combining features of mean-oscillation sparse bounds and of the cancellative sparse bounds of Conde-Alonso, Lorist and Rey. Using wavelet representations, we characterize the Calderón-Zygmund operators whose derivatives admit these bounds by explicit polynomial testing conditions. In the homogeneous setting, the same conditions characterize weighted Hardy-Sobolev inequalities with linear dependence on the $A_\infty$ characteristic. This quantitative dependence detects the transition from BMO to $L^\infty$ testing conditions that Lerner found for pointwise bounds by Riesz potentials, a distinction that qualitative weighted estimates cannot recover. In the inhomogeneous setting, we prove corresponding weighted inequalities and show that they are strictly weaker than pointwise sparse domination from the second derivative order onward. Consequences include Hardy-Sobolev estimates below the Banach range, sharp weighted Sobolev inequalities, and pointwise bounds by Riesz potentials.

math.CA↗