Papers

Problem A33 of Croft–Falconer–Guy records Reznick's question about infinite plane sets for which every five-point subset determines an ellipse or, respectively, a hyperbola, and suggests that $|x|^{2.001}+|y|^{2.001}=1$ might yield only ellipses. We give two results. First, if $p>2$ and $a>0$, every five distinct points of the power curve $u=ay^p$, $y>0$, determine a nondegenerate hyperbola; bounded subarcs therefore give nonconic rectifiable examples for the hyperbolic side of Reznick's question. Second, every fixed one-sided five-point profile, contracted toward an axial vertex of the Lamé oval $|x|^p+|y|^p=1$, eventually determines a nondegenerate hyperbola. Consequently, the suggested Lamé oval with $p=2.001$ has five-point subsets that determine nondegenerate hyperbolas, so it does not yield only ellipses. Symmetric profiles straddling the same vertex, however, determine ellipses. We also separate these results from the classical osculating-conic criterion supplied by equi-affine curvature.

An ordered positive bidiagonal factorization (PBF) is used to develop a spectral and probabilistic theory for Markov transition matrices of arbitrary finite bandwidth. The factors determine two families of mixed-type multiple orthogonal polynomials, an entrywise positive $q\times p$ matrix of measures, and a sequence of elementary death-or-stay and birth-or-stay transitions. This yields Karlin-McGregor formulas for transition probabilities, Green kernels, resolvents, potentials, and first-passage transforms without reversibility or block symmetrizability. Rational stochastic PBFs are characterized by ordered finite-urn experiments. Cyclic reorderings give Darboux intertwinings, while a factor-resolved continued fraction gives the first-return law. Grouping states produces a finite-phase quasi-birth-and-death process and a matrix continued fraction for return time and phase. For bounded continuous-time generators, uniformization preserves the spectral data. For unbounded rates, a conservative generator whose shifted leading truncations all admit scalar PBFs must be tridiagonal; nevertheless, $Q=V(T-I)$ preserves arbitrary fixed bandwidth and separates the embedded chain from the holding rates. The theory is explicit for mixed Piñeiro and Jacobi-like systems. For Piñeiro, the exact PBF region and larger structural-band positivity regions are obtained. Jacobi-like beta-convolution weights admit Gamma-factor cancellations, complete cancellation recovering Piñeiro. For $q\in\{2,3,4\}$, the strict ordering conditions give the only open PBF region, with further lower-dimensional cancellation strata. Rational $(3,2)$ examples provide all factors and the resulting Markov models.

This article constructs a Bessel family of mixed-type multiple orthogonal polynomials for a $q\times p$ matrix weight on the unit circle. Unlike a rank-one product weight, this matrix has generic rank $\min\{q,p\}$ outside a finite subset of the circle. Its reciprocal-Gamma moments recover the multiple Bessel system when $q=1$ and the Bessel-like system of Wolfs when $p=1$. The same matrix is obtained as a scaled Markov-Stieltjes limit of a rank-one Jacobi system, although the interval measures themselves have no finite limit. For balanced near-diagonal indices, explicit formulas are obtained for the mixed $A$ and $B$ polynomial vectors. Their orthogonality and weak normality are proved, and componentwise strong normality is characterized. The components have terminating generalized hypergeometric representations; the $B$ components also admit finite Kampé de Fériet representations and a matrix Rodrigues-type formula. In the one-row reduction, the bivariate representation becomes a generalized hypergeometric polynomial governed by a reflected type-II multiple Hahn polynomial. Finite Gamma-Pochhammer formulas give the near-diagonal and step-line recurrence coefficients. The corresponding banded recurrence matrix has a bidiagonal Christoffel factorization: the lower factors are evaluated from transformed polynomial vectors, while the upper factors are expressed through finite tau determinants. When $q=1$, every Christoffel step remains within the multiple Bessel family, and Gamma-Vandermonde determinants yield the complete factorization.

We study Kadec-Klee properties in spaces of homogeneous polynomials and multilinear forms by presenting optimal results. Indeed, our main tool is a family of concentration theorems showing that, on suitable $p$-convex Banach sequence lattices, a $2$-homogeneous polynomial or a bilinear form which almost attains its norm at finitely supported vectors is uniformly close to its restriction to the corresponding finite set of coordinates. As a consequence, we obtain the weak-star uniform Kadec-Klee property for spaces of complex $2$-homogeneous polynomials and complex bilinear forms on $p$-convex Banach sequence lattices with constant one for some $p>2$. This applies, in particular, to $c_0$, $\ell_p$, Lorentz spaces $d(w,p)$, Garling spaces $g(w,p)$, among others. In the real setting, a concentration argument based on the modulus of convexity of power type yields further positive results for spaces of operators and bilinear forms, extending previous results in the literature. We also prove that the restriction $p>2$ is optimal for $2$-homogeneous polynomials on $\ell_p$, $d(w,p)$ and $g(w,p)$, and establish general failures of the sequential weak-star Kadec-Klee property for real homogeneous polynomials of degree at least two and for complex homogeneous polynomials and symmetric multilinear forms of degree at least three. These results show that our positive results are optimal respect to the degree and that the assumption $p>2$ is optimal for the classical spaces $\ell_p$, $d(w,p)$ and $g(w,p)$. As a consequence of our results, we establish the strong subdifferentiability of the associated projective and symmetric projective tensor norms.

We show that the minimal degree of a Salem number with prescribed negative trace is bounded below by $T^2/\log T$ and above by $T^2\log T$, up to explicit constants and lower-order terms, where $T$ is the absolute value of the trace. This improves the previous doubly exponential upper bound of McKee and Smyth. Building on their construction, we also prove that for every prescribed negative trace there exists a constant $d_0$ such that Salem numbers of that trace exist in every even degree exceeding $d_0$.

We establish lower bounds for the canonical height of a wandering point that decays like the square of the field degree. Our methods and results apply to centered, postcritically finite, hyperbolic polynomials of prime power degree whose coefficients are algebraic integers. Our approach is ultimately inspired by an idea of Dimitrov that led to his proof of the Schinzel–Zassenhaus Conjecture. The height lower bound is derived from the lower bound of the local canonical height at an archimedean place. In previous work, the authors used a result of Dubinin on the transfinite diameter of a star-shaped tree. In this paper, we develop tools to bound the transfinite diameter of more general finite trees in the complex plane. We construct these trees using the Hubbard tree of a postcritically finite polynomial. Thurston's notion of core entropy helps us analyze combinatorial properties of the Hubbard tree.

Estimation of quantile inequality curves and measures is considered in a parametric model based on grouped data. The unknown parameters of the distribution are estimated using the minimum divergence method, using various $\phi$-divergences. The consistency of the plug-in estimators of the inequality curves and measures and the asymptotic normality of the indices estimators are proved. In a simulation study, the methods are verified and compared in terms of the accuracy of the estimation. The practical applications of the proposed methods are illustrated by the analysis of two real data sets.

Gradient-based Bayesian inference methods require efficient access to Jacobian and adjoint-Jacobian operators of high-dimensional forward models. While multigrid solvers provide near-optimal complexity for elliptic partial differential equations, they are rarely available in forms compatible with automatic differentiation (AD). We develop a differentiable multigrid solver for steady-state diffusion-absorption problems and derive its adjoint operations analytically through the full multigrid hierarchy. The resulting solver, DMGS, is implemented in C++ and interfaced with JAX to provide efficient Jacobian-vector and vector-Jacobian products for variational inference in the NIFTy framework. We validate the approach on a 3D inverse problem involving diffuse radiative transfer in tissue, reconstructing an effective radiative source from Monte Carlo-simulated data. The reconstruction reproduces the data at a reduced chi-squared of 1.1 and generalizes to 32 independent validation datasets. Benchmarks against a JAX-native multigrid implementation show comparable runtimes and consistently lower peak memory for the hand-derived adjoint, with modest reverse-mode overhead. These results establish differentiable multigrid solvers as practical building blocks for variational inference in PDE-constrained problems.

We prove that finite simple groups $G(p)$ of bounded rank and with $p$ prime have uniform expansion, that is, the family of all the Cayley graphs forms a family of (two-sided) expanders, except perhaps when $p$ belongs to a small family of exceptional primes. Furthermore, for all prime powers $q$, the set of possible exceptions to uniform expansion of $G(q)$ is shown to have “dimension zero”. We also extend these results to semisimple and perfect algebraic groups.

This paper establishes an existence and uniqueness theorem for a class of functional ordinary differential equations under assumptions weaker than those found in the literature. We then consider the problem of designing a lens that refracts rays of two distinct frequencies emitted from a point source into a common collimated beam. We derive a system of functional ordinary differential equations whose solvability characterizes the existence of such lenses. Using the existence theorem, we prove that such a lens exists in two dimensions. We then extend the corresponding result to three dimensions.

We develop a symplectic formulation of the Lagrangian Radon transform based on the transitive action of the symplectic group on the Lagrangian Grassmannian. This geometric framework naturally extends the classical notion of the marginals of the Wigner transform, viewed as elementary tomograms, to arbitrary Lagrangian subspaces. The resulting generalized tomography provides a unified treatment of position and momentum measurements related by symplectic rotations. As applications, we revisit the Pauli reconstruction problem and establish reconstruction formulas for density operators from generalized Lagrangian tomograms.

Making a progress in an Erdös problem dating back to 1957, we show that $$\max_{t \in {\Bbb R}}{|P_n(e^{it})|^2} \geq n+1 + \frac{1}{38}n^{1/3}$$ for every polynomial $P_n$ of degree $n$ with all of its coefficients in $\{-1,1\}$.

Condat's algorithm is an efficient dynamic-threshold method for projection onto the simplex, but its extension to weighted equality constraints and the algorithmic roles of resetting and removal have received limited analysis. We develop a dynamic-threshold algorithm (DTA) for a continuous quadratic knapsack problem with a weighted equality constraint. DTA maintains a threshold invariant through three operations–addition, reset, and removal–and we establish its finite termination and correctness. A sufficient condition under which reset cannot occur motivates a simpler no-reset variant, NDTA. We construct instances for which DTA runs in ${\Theta} (n)$ time whereas NDTA requires $\Theta (n^2)$ time, although both algorithms have quadratic worst-case complexity. We further show that, when the weight ratio and the number of deletions per removal pass are bounded, a linear number of passes with positive threshold increments requires the minimum nonzero gap between input values, normalized by the data range, to be at most $\exp [- \Theta (n \log n)]$. Numerical experiments with up to 10^7 variables demonstrate that DTA and NDTA achieve approximately linear empirical scaling, and outperform Secant, WMVA, Variable Fixing, Newton, Median Search, Heap, and Sort in running time.

We introduce $k$-cascading parking functions, a parametrized variant of parking functions in which cars form bumping cascades of up to $k \geq 0$ cars. Setting $k = 0$ recovers classical parking functions, whereas $k = 1$ recovers MVP parking functions. Although parking functions and cascading parking functions are equivalent as sets, they are generally distinct as maps. Therefore, in this paper we consider the enumeration of the fibers of their outcomes. Our main result is a recursive, permutation pattern-based formula for the size of the fiber of any given permutation, for any given $k \geq 0$. When specialized to the longest word, the formula reduces to a family of integer sequences that interpolate between the simplest sequence ($k=0$), the Motzkin numbers ($k = 1$), and the Catalan numbers ($k\geq n-1$). When specialized to the set of layered permutations, the formula gives new combinatorial interpretations for the row sums of certain convolution triangles, including Motzkin and Catalan convolution triangles.

The rings of non-square matrices based on the dimension-keeping (DK-) semi-tensor product (STP) are considered, where a virtual identity is introduced to make each ring possess an identity element. Using this ring structure, four kinds of eigenvalues/eigenvectors (EEs) of hypermatrices, namely, the ordinary EE (OEE), the universal EE (UEE), the diagonal EE (DEE), and the horizontal diagonal EE (HDEE), are proposed with respect to preassigned matricizings. The Kronecker canonical form (KCF) of non-square pencils is used to calculate the OEEs; the monic decomposition algorithm (MDA) is then applied to extract the UEEs, DEEs, and HDEEs from them. Finally, the KCF of non-square pencils is further used to construct the KCF of a tensor, which reveals all the EEs of the tensor. The KCF of a tensor not only shares the main properties of the Jordan canonical form of matrices but also includes the latter as a special case. Consequently, all the EEs of a hypermatrix are straightforwardly computable via its KCF.

In this paper, we investigate the low Mach number limit for the three-dimensional compressible Navier–Stokes–Korteweg equations in the whole space under a small stationary external force. We first construct a family of small stationary solutions uniformly with respect to the Mach number $\epsilon$ and prove that both the stationary density fluctuation and the compressible component of the stationary velocity are of order $\epsilon^2$. For ill-prepared non-stationary perturbations around these stationary solutions, we establish the existence and uniqueness of global strong solution by combining uniform high-order energy estimates with a low-frequency Besov estimate and a Kawashima-type compensating functional. The main difficulty is that Korteweg tensor not only changes the elliptic structure of the stationary problem, but also modifies the dispersive mechanism of the acoustic modes. In Korteweg-symmetric variables, the associated spectral projections are uniformly bounded zero-order Fourier multipliers, while the acoustic-capillary phase is wave-like at low frequencies and Schrödinger-like at high frequencies. Since the source terms generated by the stationary coefficients are generally not integrable in time, we decompose the Duhamel source according to its time-integrability and frequency behavior. Dyadic dispersive estimates, high-frequency damping estimates, and maximal regularity for the heat equation yield the global-in-time convergence rate $\epsilon^{\min\{1/r,\,1/2-1/p\}}$ in the mixed Besov norms $L^r(0,\infty;\dot B^s_{p,1})$. As a consequence, Besov embeddings also yield quantitative convergence in the mixed Lebesgue norms $L^r(0,\infty;L^p)$.

For ordinary permutations on $n$ letters, the distribution of the number of fixed points of random permutations is well known to approach the Poisson$(1)$ distribution in total variation distance as $n\to\infty$ super-exponentially quickly. We use Stein's method to get related results for the number of fixed points of random permutations of multisets. Given a sequence of multisets on $n$ letters whose expected number of fixed points converges to a constant $c$, we must have $c\geq 1$ and the distribution of number of fixed points converges to the ${\rm Poisson}(c)$ distribution as $n\to\infty$. If $c>1$, then the rate of convergence in total variation distance can be as slow as $n^{-1/2}$.

We study optimal integral representations in projective tensor products, focusing on two questions: whether they can always be replaced by countable optimal decompositions, and whether the resulting notion depends on the Borel topology used on the product of the unit balls. We show that every infinite-dimensional Banach space $X$ admits an equivalent norm for which, denoting the resulting space by $Z$, there exists an affine homeomorphic embedding \[ \Psi : \mathcal{P}([0,1]) \to S_{Z \widehat\otimes_{\pi} Z} \] such that $\Psi (\mathcal{P}([0,1]) ) \subseteq \operatorname{INA}_\pi (Z\widehat\otimes_{\pi} Z)$ and \[ \Psi (\alpha) \in \operatorname{NA}_\pi(Z \widehat\otimes_{\pi} Z) \iff \text{$\alpha$ is countably supported}. \] In particular, this implies that there exists a tensor \[ u \in \operatorname{INA}_{\pi}(Z\widehat\otimes_{\pi} Z) \setminus \operatorname{NA}_{\pi}(Z\widehat\otimes_{\pi} Z) \] and the witnessing measure may be chosen to be a nonatomic Radon probability measure. The construction realizes an affine copy of $\mathcal P([0,1])$ as an exposed face of $B_Z$ and compares the diagonal Lebesgue coupling with countable mixtures of product measures. We also prove that the norm, weak, and–on dual spaces–weak-star versions of integral projective norm attainment define the same class of tensors. Consequently, every infinite-dimensional separable reflexive Banach space $X$ with the approximation property admits an equivalent norm such that, for the resulting space $Z$, $\operatorname{NA}_{\pi}(Z\widehat\otimes_{\pi} Z) \subsetneq \operatorname{INA}_{\pi}(Z\widehat\otimes_{\pi} Z) = Z\widehat\otimes_{\pi} Z$.

In this paper, by constructing double homology for graded submanifolds of $\Delta$-manifolds, we study the double homology of hyper(di)graphs on Riemannian manifolds. With the help of the canonical projection $\pi$ from ordered sequences to their underlying unordered sets, we prove a commutative diagram of the double homology and a commutative diagram of the spaces of differential forms of hyper(di)graphs on Riemannian manifolds. Moreover, we give some relations between the fundamental group(oid)s of hyper(di)graphs on surfaces and the (pure) braid groups on surfaces by certain homomorphisms induced from $\pi$. As applications, we study the homotopy types of filtrations for packings and coverings of balls on Riemannian manifolds with hypergraph constraints, and we characterize regular maps on Riemannian manifolds by geometric realizations of the independence complexes and hypergraphs on the manifolds.

For any distinct primes $p$ and $q$, we prove that there is a finite group which does not embed into any finite group invariably generated by an element of order $p$ and an element of order $q$. This gives a negative answer to Problem 21.142 of the Kourovka Notebook \cite{kourovka21}. In fact, for every fixed pair $p,q$, the group $A_n$ cannot embed into such a group once $n$ is sufficiently large in terms of $p$ and $q$.

Reweighting source samples to match a target covariate distribution is a standard response to distribution shift when generalizing evidence from one population to another. This strategy is well suited to deterministic, learnable covariate discrepancies, but can be insufficient when source–target population differences also contain changes beyond covariate shift or when estimation of the density-ratio weights is unstable. To address this challenge, we introduce a new model that allows non-systematic changes between two population laws after systematic shifts are accounted for. Such residual shift is modeled as random perturbations to the probability space that cannot be represented in a learnable way. In this way, we separate systematic shifts, treated as bias and corrected by reweighting, from residual random perturbations, treated as distributional uncertainty and handled through dataset pooling. Under pure random perturbations, this principle yields Augmented Inverse Distance Weighting (AIDW), which uses regression augmentation and variance-optimal dataset-level pooling. For mixed shifts, we develop Augmented Inverse Hybrid Weighting (AIHW), which interpolates between AIDW and standard augmented importance weighting. Both methods trade off sampling uncertainty and distributional uncertainty via a distributional distance that describes the strength of random perturbations. We establish asymptotic properties of the methods, together with plug-in guidance for choosing tuning parameters and model diagnostic tools. Experiments on three real-world multi-site datasets demonstrate consistent reductions in mean-squared error compared with standard weighting baselines, along with substantially improved empirical coverage in settings where covariate-shift adjustment alone undercovers, showing the robustness of the proposed methods across diverse distribution shift scenarios.

For a graph $G$, let $\psi(G)=\max\{\chi(G[V(C)]):C$ is an odd cycle of $G\}$, with $\psi(G)=0$ when $G$ is bipartite. For positive integers $N$, set $F(N)=\max\{\chi(G)-\psi(G):|V(G)|\le N\}$. The function $F$ measures the finite-order additive gap arising from an open problem of Erdos and Hajnal. We prove $N^{1/6-o(1)}\le F(N)<\sqrt{6N}$. The lower bound raises the finite-order scale supplied by the Cameron-Clow path-colour construction from $\log N/\log\log N$ to a fixed power of $N$. Its proof constructs a palette-code graph from a binary covering code $\mathcal{C}\subseteq\{0,1\}^p$ and establishes the exact identities $\chi(G)=2p+\ell-\rho(\mathcal{C})$ and $\psi(G)=2p$. Near-middle Hamming coverings yield the exponent $1/6$. The upper bound combines Polavarapu's connectivity theorem, the Chvatal-Erdos Hamiltonicity theorem, and maximum-independent-set stripping.

This paper is concerned with the analysis of a one-dimensional power-law model for compressible fluid dynamics on $\mathbb{R}$, in which the shear stress takes the form $\mu |\partial_{x}u|^{p-2}\partial_{x}u$, where $\mu$ is the viscosity coefficient and $u$ is the velocity. We prove that, in the singular limit $p\rightarrow\infty$, the solutions converge to functions $(\rho,u)$ satisfying $|\partial_{x}u|\leq 1$, $\tau = \pi \partial_{x}u$, $\pi \geq 0$, and $\pi (1 - |\partial_{x}u|) = 0$ a.e. on $\mathbb{R}$. Moreover, we rigorously justify the existence of weak solutions to the limiting equation. The convergence as $p \to \infty$ is obtained via domain truncation and compactness arguments, of which the key challenge is to show that the density remains bounded away from zero and infinity on any compact subset. This extends the recent result of Bresch, Burtea, and Szlenk [Nonlinearity 26 (2026), no. 5, Paper No. 055010.] from one-dimensional periodic domain to the whole real line.

We estimate convergence rates of value functions for particle approximations of a controlled nonlinear filtering problem. The state is a McKean–Vlasov diffusion on the flat torus, driven by hidden idiosyncratic noise and observed common noise. The filter—the conditional law of the state given the observations—serves as the state variable of the control problem, and the associated value function solves a second-order Hamilton–Jacobi–Bellman equation on the Wasserstein space. We approximate this problem by a centralized \(N\)-particle control problem with independent idiosyncratic noises and a common observation noise. The framework accommodates nonseparable rewards and controlled drifts. Since a single control is applied to the entire population, the Hamiltonian is defined by an optimization performed after integration over the population. Under smoothness of the data, uniform ellipticity, and regularity of this Hamiltonian, we establish uniform value-function error bounds of order \(N^{-1/6}\) for \(d=1\), \(N^{-1/6}(\log N)^{1/3}\) for \(d=2\), and \(N^{-1/(3d)}\) for \(d>2\). The proof combines a translation lift in the common-noise direction, Fourier–Wasserstein inf- and sup-convolutions, viscosity comparison, and particle derivative estimates uniform in \(N\).

The class of six-dimensional totally nondegenerate CR submanifolds in $\mathbb C^5$ contains an infinite number of models parameterized by nonzero pairs $(a, b)\in\mathbb C\times\mathbb R$ appearing in their defining equations. Employing the equivariant moving frame method, we construct normal forms of this class. The applied normalizations yield into fourteen biholomorphically inequivalent subclasses, each with its own normal form. For each subclass, we determine the Lie algebra of infinitesimal CR automorphisms. Finally, using the method of involution, we prove the convergence of the constructed normal forms.

A reciprocal transformation is applied to link an integrable extension of the classical solitonic mKdV to a novel nonlinear evolution equation incorporating a source term. Application of Ermakov-Painleve' II symmetry reduction is made to determine the exact solution to a class of associated moving boundary problems of Stefan-type.

Cowreaths of type $(A \otimes H^{\mathrm{op}},H,\psi)$ have been investigated in [10,19-21,27] as examples of (h-)separable and Frobenius coalgebras in monoidal categories. They are entwining structures built from a Hopf algebra $H$ and an $H$-comodule algebra $(A,\rho_A)$. In this article we develop a general theory that allows us to recover results from the aforementioned papers in an easier way and also to extend them to cowreaths in higher dimension. Focusing on separability, the crucial observation is that, when $H$ has bijective antipode, one should work with integrals on the coalgebra $(H,\psi)$ in place of Casimir morphisms, for they are easier to classify. On the other hand, in dealing with Frobenius properties, we obtain that a cowreath $(A \otimes H^{\mathrm{op}},H,\psi)$ is Frobenius if and only if the morphism $(\mathrm{Id}_A \otimes \mu)\rho_A$ is inner ($\mu$ is the distinguished grouplike element in $H^*$). While Frobenius cowreaths are always (h-)separable, examples of (h-)separable cowreaths that are not Frobenius will be presented at the end of this article.

We give a deterministic algorithm for finding all integer roots of a square-free polynomial $f\in\mathbb Z[x]$ of degree $n$ with $\lVert f\rVert_\infty<2^b$. The running time is $$ \tilde{O}(n^{3/2}b), $$ improving the $\tilde{O}(n^2b)$ bound of Harvey and Hittmeir (Research in Number Theory, 2022). The algorithm follows the classical $p$-adic framework: find roots modulo a prime $p$, lift them modulo a high power of $p$, and verify the lifted candidates. The main new idea is to avoid searching for a prime for which $f\bmod p$ is square-free. Instead, we find a prime for which the total multiplicity of repeated roots modulo $p$ is small. This requires lifting repeated roots, which we handle using a weighted lifting tree. We also give a faster deterministic candidate-verification algorithm: given $n$ candidate integers smaller in absolute value than $2^b$, we decide which are roots of $f$ in $$ \tilde{O}(nb+\min(n^2,nb^2)) $$ bit operations. Together, these ingredients give the first deterministic subquadratic-in-$n$ improvement for integer root finding in the square-free case.

We provide a new proof, model-theoretic and geometric in nature, of the tame henselian rationality theorem of Kuhlmann. The proof takes place within the framework of stable completions of algebraic varieties over a valued field. It relies on the proof of two main results of independent interest: a Nash structure theorem for the stable completion of an algebraic curve and a tame descent theorem for abstract polydiscs.

Optimal transport potentials are fundamental objects in statistics, economics, and machine learning: their gradients generate optimal transport maps, while the potentials themselves act as location-dependent dual prices and sensitivity variables. We study the estimation of the quadratic optimal transport potential when a fixed absolutely continuous reference distribution $\mu$ is transported to an unknown distribution $\nu$, accessed to via its empirical measure. Our main ingredient is a stability inequality that controls the $L^1(\mu)$ distance, modulo additive constants, between a strongly convex potential $\varphi$ and a convex potential $\widetilde\varphi$ by a weak dual norm of $(\nabla\widetilde\varphi)_\#\mu-(\nabla\varphi)_\#\mu,$ together with a second-order Wasserstein remainder of logarithmic type. This separation between the leading empirical-process term and the Wasserstein remainder yields faster convergence for potentials than for the corresponding transport maps. Under smoothness and uniform convexity assumptions, the exact semidiscrete Brenier potential converges in $L^1(\mu)$ at rate $n^{-1/2}$ for $d\leq3$, at rate $n^{-1/2}(\log n)^{5/2}$ for $d=4$, and at rate $n^{-2/d}(\log n)^{(d+2)/d}$ for $d\geq5$. The polynomial exponents are sharp. In dimensions $d\leq3$, we further establish a nondegenerate function-space central limit theorem and prove consistency of the nonparametric bootstrap. These results yield joint root-$n$ inference for every fixed finite collection of normalization-invariant weighted contrasts of the potential, including regional shadow premia in reference-based risk problems. Finally, we prove matching upper and lower bounds of order $\varepsilon\log(1/\varepsilon)$ for the normalization-invariant sum of the entropic dual potentials.