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Apoorva Khare

Publications and source records attributed to Apoorva Khare.

At least 19 recordsLinked to original sources
arXiv2026

A tight bound for affine-linearity, via universal ballot matrices

Based on work with Greenfeld and with Ziegler, Tao showed a concatenation result that if a map $f : \mathbb{F}^2 \to \mathbb{F}$ is affine-linear on every line parallel to the coordinate axes, and on all lines with a fixed nonzero slope (where the field $\mathbb{F}$ has size $> 2$), then $f$ is affine-linear on $\mathbb{F}^2$. We extend this from $\mathbb{F}^2$ to $\mathbb{F}^n$ and obtain a tight minimum number of additional lines needed -- $N = \binom{n}{\lfloor n/2 \rfloor}$ -- for every field $\mathbb{F}$ with $3 \leqslant n < |\mathbb{F}|$. The proof is constructive and shows a stronger result: the existence of a universal family of $0$-$1$ matrices of size $\binom{n}{k} \times \binom{n}{k}$ (one for each pair $0 \leqslant k \leqslant n$), which are indexed by ballot sets and are unimodular over all unital commutative rings. We also show a second tightness: of the assumption $n < |\mathbb{F}|$. Else, there exist multi-affine maps $f$ which are affine-linear on every line through the origin, but not affine-linear globally on $\mathbb{F}^n$. More strongly, we prove this dichotomy -- including the bound of $N$ -- over all integral domains, or Noetherian (e.g.\ finite or Artinian) rings, or products of these. This yields a novel numerical invariant for affine-linearity, for every product of Noetherian rings and integral domains.

math.CO↗
arXiv2026

Distance preservers for Lobachevsky space

We obtain a complete description of the class of entrywise preservers of Lorentz-Gram matrices. This resolves, for the case of constant negative curvature, the classification of entrywise preservers obtained by Schoenberg in the zero-curvature (Euclidean) and constant-positive-curvature (spherical) settings.

math.CA↗
arXiv2026

The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings

We present an overview of a classical theme in analysis and matrix positivity: the question of which functions preserve positive semidefiniteness when applied entrywise. In addition to drawing the attention of experts such as Schoenberg, Rudin, and Loewner, the subject has attracted renewed attention owing to its connections to various applied fields and techniques. In this survey we will focus mainly on the question of preserving positivity in all dimensions. Connections to distance geometry and metric embeddings, positive definite sequences and functions, Fourier analysis, applications and covariance estimation, Schur polynomials, and finite fields will be discussed. The Appendix contains a mini-survey of sphere packings, kissing numbers, and their "lattice" versions. This part overlaps with the rest of the article via Schoenberg's classification of the positive definite functions on spheres, aka dimension-free entrywise positivity preservers with a rank constraint - applied via Delsarte's linear programming method.

math.CA↗
arXiv2026

Negativity-preserving transforms of tuples of symmetric matrices

Compared to the entrywise transforms which preserve positive semidefiniteness, those leaving invariant the inertia of symmetric matrices reveal a surprising rigidity. We first obtain the classification of negativity preservers by combining recent advances in matrix analysis with some novel arguments relying on well chosen test matrices, Sidon sets from number theory, and analytic properties of absolutely monotone functions. We continue with the analogous classification in the multi-variable setting, revealing for the first time a striking separation of variables, with absolute monotonicity on one side and only homotheties on the other. We conclude with the complex analogue of this result.

math.CA↗
arXiv2026

Majorization via positivity of Jack and Macdonald polynomial differences

Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021), McSwiggen--Novak (2022), and Chen--Sahi (2024+) among others. Here we extend the inequalities by these authors to Jack and Macdonald polynomials, and obtain conjectural characterizations of majorization and of weak majorization of the underlying partitions. We prove these characterizations for two variables. In fact, we upgrade -- and prove in the above cases -- the characterization of majorization, to containment of Jack and Macdonald differences lying in the Muirhead semiring.

math.CO↗
arXiv2025

Maps preserving the sum-to-difference ratio

For a field $\mathbb{F}$, what are all functions $f \colon \mathbb{F} \rightarrow \mathbb{F}$ that satisfy the functional equation $f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$ for all $ x \neq y$ in $\mathbb{F}$? We solve this problem for the fields $\mathbb{Q}, \mathbb{R}$, and a class of its subfields that includes the real constructible numbers, the real algebraic numbers, and all quadratic number fields. We also solve it over the complex numbers and on any subfield of $\mathbb{R}$, if $f$ is continuous over the reals. The proofs involve a mix of algebra in all fields, analysis over the real line, and some topology in the complex plane.

math.NT↗
arXiv2025

Numerical radius and $\ell_p$ operator norm of Kronecker products and Schur powers: inequalities and equalities

Suppose $A=[a_{ij}]\in \mathcal{M}_n(\mathbb{C})$ is a complex $n \times n$ matrix and $B\in \mathcal{B}(\mathcal{H})$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$. We show that $w(A\otimes B)\leq w(C),$ where $w(\cdot)$ denotes the numerical radius and $C=[c_{ij}]$ with $c_{ij}= w\left(\begin{bmatrix} 0& a_{ij}\\ a_{ji}&0 \end{bmatrix} \otimes B\right).$ This refines Holbrook's classical bound $w(A\otimes B)\leq w(A) \|B\|$ [J. Reine Angew. Math. 1969], when all entries of $A$ are non-negative. If moreover $a_{ii}\neq 0$ $ \forall i$, we prove that $w(A\otimes B)= w(A) \|B\|$ if and only if $w(B)=\|B\|.$ We then extend these and other results to the more general setting of semi-Hilbertian spaces induced by a positive operator. In the reverse direction, we also specialize these results to Kronecker products and hence to Schur/entrywise products, of matrices: (1)(a) We first provide an alternate proof (using $w(A)$) of a result of Goldberg-Zwas [Linear Algebra Appl. 1974] that if the spectral norm of $A$ equals its spectral radius, then each Jordan block for each maximum-modulus eigenvalue must be $1 \times 1$ ("partial diagonalizability"). (b) Using our approach, we further show given $m \geq 1$ that $w(A^{\circ m})\leq w^m(A)$ - we also characterize when equality holds here. (2) We provide upper and lower bounds for the $\ell_p$ operator norm and the numerical radius of $A\otimes B$ for all $A \in \mathcal{M}_n(\mathbb{C})$, which become equal when restricted to doubly stochastic matrices $A$. Finally, using these bounds we obtain an improved estimation for the roots of an arbitrary complex polynomial.

math.FA↗
arXiv2025

Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions

In 2022, Huh-Matherne-Mészáros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type $A$ root directions). A reinterpretation of this result is that the characters of finite-dimensional simple representations of $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian (DL). In the same paper, these authors also showed that shifted characters of Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are DL. In this work we extend these results to a larger family of modules that subsumes both of the above: we show that shifted characters of all parabolic Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. The proof involves certain graphs on $[n+1]$; more strongly, we explain why the character (i.e., generating function) of the Kostant partition function of any loopless multigraph on $[n+1]$ is Lorentzian after shifting and normalizing. We then show that parabolic Vermas form a "maximal" class with log-concave (hence DL) characters. Namely, log-concavity fails in greater generality along three natural directions: (1) it does not hold for every simple Lie type, (2) nor for a larger universal family of highest weight modules, the higher order Verma modules, even in type $A$, and (3) it does not always hold for important generalizations of Schur polynomials: the Jack and Macdonald polynomials. Finally, we extend these results to parabolic (i.e. "first order") and higher order Verma modules over the semisimple Lie algebras $\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$. We also partially resolve a conjecture of Huh et al on the DL property for integral highest weight simple modules.

math.RT↗
arXiv2025

Cholesky decomposition for symmetric matrices, Riemannian geometry, and random matrices

For each $n \geq 1$ and sign pattern $ε\in \{ \pm 1 \}^n$, we introduce a cone of real symmetric matrices $LPM_n(ε)$: those with leading principal $k \times k$ minors of signs $ε_k$. These cones are pairwise disjoint and their union $LPM_n$ is an open dense cone in all symmetric matrices; they subsume positive and negative definite matrices, and symmetric (P-,) N-, PN-, almost P-, and almost N- matrices. We show that each $LPM_n$ matrix $A$ admits an uncountable family of Cholesky-type factorizations - yielding a unique lower triangular matrix $L$ with positive diagonals - with additional attractive properties: (i) each such factorization is algorithmic; and (ii) each such Cholesky map $A \mapsto L$ is a smooth diffeomorphism from $LPM_n(ε)$ onto an open Euclidean ball. We then show that (iii) the (diffeomorphic) balls $LPM_n(ε)$ are isometric Riemannian manifolds as well as isomorphic abelian Lie groups, each equipped with a translation-invariant Riemannian metric (and hence Riemannian means/barycentres). Moreover, (iv) this abelian metric group structure on each $LPM_n(ε)$ - and hence the log-Cholesky metric on Cholesky space - yields an isometric isomorphism onto a finite-dimensional Euclidean space. The complex version of this also holds. In the latter part, we show that the abelian group $PD_n$ of positive definite matrices, with its bi-invariant log-Cholesky metric, is precisely the identity-component of a larger group with an alternate metric: the open dense cone $LPM_n$. This also holds for Hermitian matrices over several subfields $\mathbb{F} \subseteq \mathbb{C}$. As a result, (v) the groups $LPM_n^{\mathbb{F}}$ and $LPM_\infty^{\mathbb{F}}$ admit a rich probability theory, and the cones $LPM_n(ε), TPM_n(ε)$ admit Wishart densities with signed Bartlett decompositions.

math.RA↗
arXiv2025

A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$

Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra $\mathfrak{h}$. Also fix a weight $λ\in\mathfrak{h}^*$. For $M(λ)\twoheadrightarrow V$ an arbitrary highest weight $\mathfrak{g}$-module, we provide a cancellation-free, non-recursive formula for the weights of $V$. This is novel even in finite type, and is obtained from $λ$ and a collection $\mathcal{H}=\mathcal{H}_V$ of independent sets in the Dynkin diagram of $\mathfrak{g}$ that are associated to $V$. Our proofs use and reveal a finite family (for each $λ$) of "higher order Verma modules" $\mathbb{M}(λ,\mathcal{H})$ - these are all of the universal modules for weight-considerations. They (i) generalize and subsume parabolic Verma modules $M(λ,J)$, and (ii) have pairwise distinct weight-sets, which exhaust the weight-sets of all modules $M(λ)\twoheadrightarrow V$. As an application, we explain the sense in which the modules $M(λ)$ of Verma and $M(λ,J_V)$ of Lepowsky are respectively the zeroth and first order upper-approximations of every $V$, and continue to higher order upper-approximations $\mathbb{M}_k(λ,\mathcal{H}_V)$ (and to lower-approximations). We determine every $k$th order integrability of $V$. We then introduce the category $\mathcal{O}^\mathcal{H}\subset\mathcal{O}$, which is a higher order parabolic analogue that contains the higher order Verma modules $\mathbb{M}(λ,\mathcal{H})$. We show that $\mathcal{O}^\mathcal{H}$ has enough projectives, and also initiate the study of BGG reciprocity, by proving it for all $\mathcal{O}^\mathcal{H}$ over $\mathfrak{g}=\mathfrak{sl}_2^{\oplus n}$. Finally, we provide a BGG resolution for our universal modules $\mathbb{M}(λ,\mathcal{H})$ in certain cases including all rank-3 $\mathfrak{g}$; this yields their Weyl-type character formulas, with the actions of parabolic Weyl semigroups.

math.RT↗
arXiv2025

Maps preserving the sum-to-difference ratio in characteristic $p$

Given a field $\mathbb{F}$, we introduce a novel group $SD(\mathbb{F})$ of its self-maps: the solutions $f \colon \mathbb{F} \twoheadrightarrow \mathbb{F}$ to the functional equation $f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$ for all $ x \neq y$ in $\mathbb{F}$. We compute this group for all fields algebraic over $\mathbb{F}_p$. In particular, this group distinguishes $\mathbb{F}_5$ among all finite fields $\mathbb{F}_q$, and in fact among all subfields of $\overline{\mathbb{F}_q}$.

math.NT↗
arXiv2024

Multivariate transforms of total positivity

Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified the post-composition operators that preserve TP/TN kernels of each specified order. We explain how to extend this from preservers to transforms, and from one to several variables. Namely, given arbitrary nonempty totally ordered sets $X,Y$, we characterize the transforms that send each tuple of kernels on $X \times Y$ that are TP/TN of orders $k_1, \dots, k_p$, to a TP/TN kernel of order $l$, for arbitrary positive integers (or infinite) $k_j$ and $l$. An interesting feature is that to preserve TP (or TN) of order $2$, the preservers are products of individual power (or Heaviside) functions in each variable; but for all higher orders, the preservers are powers in a single variable. We also classify the multivariate transforms of symmetric TP/TN kernels; in this case it is the preservers of TP/TN of order 3 that are multivariate products of power functions, and of order 4 that are individual powers. The proofs use generalized Vandermonde kernels, Hankel kernels, (strictly totally positive) Polya frequency functions, and a kernel studied recently but tracing back to works of Schoenberg [Ann. of Math. 1955] and Karlin [Trans. Amer. Math. Soc. 1964].

math.FA↗
arXiv2024

Preserving positivity for rank-constrained matrices

Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg [Duke Math. J. 9, 1942] and Rudin [Duke Math. J. 26, 1959]. Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $n$. Such characterizations for a fixed value of $n$ are difficult to obtain, and in fact are only known in the $2 \times 2$ case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping $n \times n$ positive semidefinite matrices of rank at most $l$ into positive semidefinite matrices of rank at most $k$ for $1 \leq l \leq n$ and $1 \leq k < n$. We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner [Trans. Amer. Math. Soc. 136, 1969] can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.

math.FA↗
arXiv2023

The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra

To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X(\{ n_x : x \in X \})$. This is obtained from the blowup $X[{\bf n}]$ - which contains $n_x$ copies of each point $x$ - by computing the determinant of the distance matrix of $X[{\bf n}]$ and removing an exponential factor. We prove that as a function of the sizes $n_x$, $p_X({\bf n})$ is a polynomial, is multi-affine, and is real-stable. This naturally associates a hitherto unstudied delta-matroid to each metric space $X$; we produce another novel delta-matroid for each tree, which interestingly does not generalize to all graphs. We next specialize to the case of $X = G$ a connected unweighted graph - so $p_G$ is "partially symmetric" in $\{ n_v : v \in V(G) \}$ - and show three further results: (a) We show that the polynomial $p_G$ is indeed a graph invariant, in that $p_G$ and its symmetries recover the graph $G$ and its isometries, respectively. (b) We show that the univariate specialization $u_G(x) := p_G(x,\dots,x)$ is a transform of the characteristic polynomial of the distance matrix $D_G$; this connects the blowup-polynomial of $G$ to the well-studied "distance spectrum" of $G$. (c) We obtain a novel characterization of complete multipartite graphs, as precisely those for which the "homogenization at $-1$" of $p_G({\bf n})$ is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of $p_G(-{\bf n})$ is strongly Rayleigh.

math.MG↗
arXiv2023

Matrix positivity preservers in fixed dimension. II: positive definiteness and strict monotonicity of Schur function ratios

We continue the study of real polynomials acting entrywise on matrices of fixed dimension to preserve positive semidefiniteness, together with the related analysis of order properties of Schur polynomials. Previous work has shown that, given a real polynomial with positive coefficients that is perturbed by adding a higher-degree monomial, there exists a negative lower bound for the coefficient of the perturbation which characterizes when the perturbed polynomial remains positivity preserving. We show here that, if the perturbation coefficient is strictly greater than this bound then the transformed matrix becomes positive definite given a simple genericity condition that can be readily verified. We identity a slightly stronger genericity condition that ensures positive definiteness occurs at the boundary. The analysis is complemented by computing the rank of the transformed matrix in terms of the location of the original matrix in a Schubert cell-type stratification that we have introduced and explored previously. The proofs require enhancing to strictness a Schur monotonicity result of Khare and Tao, to show that the ratio of Schur polynomials is strictly increasing along each coordinate on the positive orthant and non-decreasing on its closure whenever the defining tuples satisfy a coordinate-wise domination condition.

math.CA↗
arXiv2023

The additive-multiplicative distance matrix of a graph, and a novel third invariant

Graham showed with Pollak and Hoffman-Hosoya that for any directed graph $G$ with strong blocks $G_e$, the determinant $\det(D_G)$ and cofactor-sum $cof(D_G)$ of the distance matrix $D_G$ can be computed from the same quantities for the blocks $G_e$. This was extended to trees - and in our recent work to any graph - with multiplicative and $q$-distance matrices. For trees, we went further and unified all previous variants with weights in a unital commutative ring, into a distance matrix with additive and multiplicative edge-data. In this work: (1) We introduce the additive-multiplicative distance matrix $D_G$ of every strongly connected graph $G$, using what we term the additive-multiplicative block-datum $\mathcal{G}$. This subsumes the previously studied additive, multiplicative, and $q$-distances for all graphs. (2) We introduce an invariant $κ(D_G)$ that seems novel to date, and use it to show "master" Graham-Hoffman-Hosoya (GHH) identities, which express $\det(D_G), cof(D_G)$ in terms of the blocks $G_e$. We show how these imply all previous variants. (3) We show $\det(.), cof(.), κ(.)$ depend only on the block-data for not just $D_G$, but also several minors of $D_G$. This was not studied in any setting to date; we show it in the "most general" additive-multiplicative setting, hence in all known settings. (4) We compute $D_G^{-1}$ in closed-form; this specializes to all known variants. In particular, we recover our previous formula for $D_T^{-1}$ for additive-multiplicative trees (which itself specializes to a result of Graham-Lovasz and answers a 2006 question of Bapat-Lal-Pati.) (5) We also show that not the Laplacian, but a closely related matrix is the "correct" one to use in $D_G^{-1}$ - for the most general additive-multiplicative matrix $D_G$ of each $G$. As examples, we compute in closed form $\det(D_G), cof(D_G), κ(D_G), D_G^{-1}$ for hypertrees.

math.CO↗
arXiv2023

Distance matrices of a tree: two more invariants, and in a unified framework

Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on the edge-data. We introduce a general framework for bi-directed weighted trees, with threefold significance. First, we improve on state-of-the-art for all known variants, even in the classical Graham-Pollak case: we delete arbitrary pendant nodes (and more general subsets) from the rows/columns of $D$, and show these minors do not depend on the tree-structure. Second, our setting unifies all known variants (with entries in a commutative ring). We further compute $D^{-1}$ in closed form, extending a result of Graham-Lovasz [Adv. Math. 1978] and answering an open question of Bapat-Lal-Pati [Lin. Alg. Appl. 2006] in greater generality. Third, we compute a second function of the matrix $D$: the sum of all its cofactors, cof$(D)$. This was worked out in the simplest setting by Graham-Hoffman-Hosoya (1978), but is relatively unexplored for other variants. We prove a stronger result, in our general setting, by computing cof$(.)$ for minors as above, and showing these too depend only on the edge-data. Finally, we show our setting is the "most general possible", in that with more freedom in the edgeweights, det$(D)$ and cof$(D)$ depend on the tree structure. In a sense, this completes the study of the invariants det$(D_T)$, cof$(D_T)$ for trees $T$ with edge-data in a commutative ring. Moreover: for a bi-directed graph $G$ we prove multiplicative Graham-Hoffman-Hosoya type formulas for det$(D_G)$, cof$(D_G)$, $D_G^{-1}$. We then show how this subsumes their 1978 result. The final section introduces and computes a third, novel invariant for trees and a Graham-Hoffman-Hosoya type result for our "most general" distance matrix $D_T$.

math.CO↗