Papers

It is shown that all solutions of the canonical equation y^a = 1 + a*x*y^b with arbitrary complex exponents a, b (case m = 1) and of its limiting cases - the binomial root y = (1 + a*x)^(1/a) at b = 0, the exponential y = e^x at a = 0, b = 0, and the Lambert equation y = e^(x*y^b) at a = 0 - as well as the logarithms of these solutions (case m = 0), are expressed by a single power master series M(m; a; b; x) = m + x + ((m - a + 2b)/2!) x^2 + ((m - a + 3b)(m - 2a + 3b)/3!) x^3 + ... For multivalued solutions beyond the radius of convergence, a geometric criterion for selecting the conjugate series (the same master series, but with a different set of parameters M(m; a'; b'; x')) is proposed, ensuring branch continuity on the complex plane. This representation of power series in terms of master numbers not only allows one to compute coefficients in degenerate cases where the gamma-function representation loses meaning, but also provides a transparent analytical algorithm for the merge operation on master series. The method is illustrated on problems in gas dynamics and plasma physics, where it is demonstrated that physically realizable states require the application of conjugate series beyond the radius of convergence of the basic expansion.

Weighted split conformal prediction reweights calibration scores by the likelihood ratio between the test and training covariate distributions and guarantees marginal coverage under covariate shift. We study its coverage conditional on the calibration data. An elementary argument, based on a single concentration inequality at a fixed population quantile, gives explicit training-conditional bounds without unspecified constants, and shows that the relevant scale is not the supremum of the likelihood ratio but a variance proxy built from the chi-squared divergence of the shift and from the average of the ratio over the part of the test population, of probability equal to the miscoverage level, where it is largest. A two-point lower bound shows that the root-m rate and the chi-squared contribution are intrinsic to the shift. Run at an explicitly inflated level, the weighted quantile becomes a deterministic PAC prediction set. We compare it with randomized rejection sampling and with importance-weighted learn-then-test and, through a certified choice of a clipping level for the likelihood ratio, map the regime in which each gives the narrower valid set. The analysis extends to estimated likelihood ratios and to tail functionals estimated from an unlabeled source sample, which yields a fully finite-sample certificate.

We present DAOCP, a dual active set solver for linear quadratic optimal control problems with stage-wise equality and inequality constraints. Active set methods are leading Model Predictive Control benchmarks for full-body robotics, but existing solvers operate on dense QPs, while typical problem dimensions favor methods that exploit the optimal control structure. DAOCP combines the warm-starting capabilities of active set algorithms with the better computational scaling of structure exploiting solvers, by relying on a generalization of the relationship between the Riccati recursion and the Cholesky factorization of the condensed Hessian. This result enables dual active set iterations to operate directly on the original optimal control problem through recursive computations, without explicitly forming the condensed quadratic program. On robotics benchmarks, DAOCP is the fastest of four solvers in four of five scenarios, cutting average solve time on a 58-state Atlas model by 9$\times$ relative to state of the art solvers DAQP and HPIPM.

We prove Landau-Ginzburg mirror symmetry conjecture for ADE singularities with the nontrivial symmetry groups. In particular, we consider $(f,G)$ with $f \in \mathbb{C}[z_1,z_2,z_3]$ defining ADE singularity and $G$ — nontrivial group of diagonal symmetries of $f$. We show that orbifold Saito theory of $(f,G)$ is isomorphic to the Fan-Jarvis-Ruan-Witten theory of its Berglund-Hübsch-Henningson dual pair $(\widetilde f, \widetilde G)$.

Locally stationary Markov chains have attracted a lot of attention in statistics in the past three decades. In this paper we prove a variety of limit theorems for partial sums generated by such chains. We first provide explicit formulas for the asymptotic mean and variance (and also higher moments), and obtain optimal convergence rates towards them. We then prove a Berry–Esseen theorem, a local central limit theorem, Edgeworth expansions, and large and moderate deviations principles. Our approach involves a parametric Perron–Frobenius theorem, which is proved using the theory of complex (Hilbert) projective metrics developed in \cite{Rugh,Dubois}, together with local approximation arguments and ideas in \cite{dolgopyat2023berry}.

Let $\Delta_u(d,n)$ be the maximum graph diameter of a pointed $d$-dimensional polyhedron with $n$ facets. Using a curved-path encoding of the iterated Kalai–Kleitman recurrence, we prove, uniformly over $n\geq d\geq 4$, \[ \Delta_u(d,n) \leq (n-d)^{\log_2 G_d},\qquad G_d = (4\ln 2+o(1))\frac{d}{(\ln d)^2}, \] where the asymptotic expression for $G_d$ is understood as $d\to\infty$, improving the exponent of the previous best quasi-polynomial bound by an additional logarithmic factor. With the quantitative $d$-step reduction, we also obtain the complementary excess-based bound \[ \Delta_u(d,n) \leq (n-d)^{\frac{1}{2}\log_2(n-d)+O(1)}, \] where the implied constant is absolute. In the regime $n - d = \Theta(d)$, the latter bound is asymptotically stronger and halves the leading coefficient in the exponent. We further examine their behavior as $n$ grows relative to $d$, obtaining sharper exponents when $n = d^{1/\gamma+o(1)}$ for fixed $0<\gamma<1$ and an almost-linear bound in the deep-tail regime $(\ln n)/d\to\infty$. We also show that the leading term of the general bound is sharp within this positive path-counting framework.

We study the sequence $a(0)=0$, $a(1)=1$ and $a(n)=n-a(n-a(n-a(n-1)))$ for $n\ge 2$, listed as A076502 in the On-Line Encyclopedia of Integer Sequences. We identify $a(n)$ as a two-position shift in the greedy Padovan numeration system, with a finite-state correction. The proof constructs an addition automaton from an exact integer-carry invariant and certifies its completeness by finite-language inclusion; a synchronized automaton then verifies the nested recurrence. We establish bounded discrepancy from the line of slope $c$, where $c^3-c^2+2c-1=0$, and show that the exact set of offsets from $\lfloor cn\rfloor$ is $\{-1,0,1,2\}$. We construct an explicit 26-letter non-erasing morphic presentation of the first-difference word, prove that its least balance constant is 4, and give an effective procedure for enclosing the global discrepancy extrema to arbitrary accuracy. We formalize the recurrence identification, six-decimal discrepancy bound, exact offset set, concrete morphic identity, least balance constant, and an effective extrema algorithm in Lean. Separate exact computations refine the numerical enclosures.

Recovering probability measures from their moments has numerous applications, esp. in connection with the method of moments in statistics and optimization. In the setting where measure need not be finitely atomic, but its support is known to be compact and semialgebraic with codimension at least one, the problem is still open. We combine moment-matrix kernel information with the Christoffel–Darboux kernel to provide a discrete approximation of the support. To validate the proposed approach, we test our algorithm on analytically computed moments and pseudo-moments arising from polynomial optimization problems without unique global minimizers. This complements well-known recent work on recovery of measures with algebraic support, where the kernel of a moment matrix can reveal polynomials vanishing on the support, and on recovery of sufficiently regular full-dimensional supports, where estimators constructed by thresholding the Christoffel–Darboux kernel are known to converge asymptotically to the support.

We propose a new $L_2$-type test for white noise which allows the dimension $p$ of the time series to either (i) be a fixed constant, or (ii) diverge with the sample size $n$. The proposed test statistic exhibits an interesting phase transition, following two different regimes of behavior: $p$ is fixed, and $p\rightarrow\infty$. Because identification of the operable regime is difficult, if not impossible in practice, we devise a novel adaptive bootstrap method to construct unified testing procedure across different phases. Numerical experiments confirm the good finite sample performance of the proposed adaptive $L_2$-type test in comparison to the existing methods in the literature. The proposed testing procedure has been implemented in R package HDTSA.

We develop an accelerated stochastic method for convex $(H_0,H_1)$-smooth optimization under heavy-tailed noise. The unbiased oracle has a finite $p$-th noise moment, $1<p\le2$, with constant, gradient-dependent, and gap-dependent terms. Our method combines accelerated updates with clipping, projection, and phase restarts. Each iteration uses a single stochastic-gradient sample, without minibatching. We prove high-probability convergence to any desired accuracy despite clipping bias. The deterministic terms retain square-root dependence on both $H_0$ and $H_1$. In our three-component noise model, strong convexity leaves a polynomial accuracy cost only for the constant component; without it, the dependence is logarithmic even for $p<2$. More generally, for noise moments scaling as $(f(x)-f^\star)^\alpha$, the stochastic cost becomes logarithmic at $\alpha=p$ for convex objectives and $\alpha=p/2$ under strong convexity.

In Integrated Sensing and Communications (ISAC), radar sensing must operate under chirp subsampling with up to 90% missing data. An attention-based baseline, limited to a 52~ms buffer, collapses toward maximum uniform entropy ($H=2.584$ bits) as sparsity increases, failing to capture long-range gait-cycle context. We propose StarBOA, which replaces attention with a causal Mamba state-space model that updates incrementally on a per-window basis without re-scanning past reconstructions. By maintaining a persistent state, StarBOA integrates over $100\times$ more temporal history at no additional per-step computational cost. StarBOA outperforms the baseline's published results across all sparsity levels, with SSIM gains increasing from $+0.0379$ at 50% missing data to $+0.2472$ at 90%. Each window is processed in 1.53~ms with zero lookahead, demonstrating efficient causal reconstruction under extreme chirp subsampling.

In this work, we propose several novel numerical schemes that synergistically integrates the $L1$ method and symmetric fractional-order reduction (SFOR) method, specifically designed for fractional sub-diffusion equations with time delay. Crucially, we rigorously establish an optimal convergence order of $2 - \alpha/2$ in temporal direction, which improves upon the classical rate of $2-\alpha$ under identical smoothness conditions. To further enhance computational efficiency, we develop a weighted alternating direction implicit (ADI) scheme for the two-dimensional case and analyze its coupling with compact spatial discretization—thereby achieving high-order accuracy in space. Moreover, we introduce a new framework for analyzing the error of $L1$-type discretization on graded temporal meshes, which relaxes the mesh grading requirement and still guarantees optimal convergence rate at continuous time points. Some numerical experiments comprehensively validate both the theoretical accuracy and computational efficiency of the resulting schemes.

In this contribution we consider the Fokker-Planck equation for fermions, commonly known as the Fermi-Dirac-Fokker-Planck equation. This equation is a correction of the classical Fokker-Planck model that incorporates the Pauli exclusion principle. We present a novel approach that combines three ingredients: (1) stability of solutions, (2) quantitative convergence for a dense set of initial data via a robust entropy method, and (3) a De Giorgi level-set iteration, to establish quantitative time-asymptotic thermalisation for general initial data. In particular, we prove quantitative convergence for arbitrarily cold initial data.

This article is devoted to the inverse problem of uniquely determining different classes of coefficients in a hyperbolic equation with space-time nonlocal attenuation from a local source-to-solution map. Our objective is to exploit the nonlocal attenuation mechanism to address inverse coefficient problems that remain open, or are even fundamentally intractable, for linear and nonlinear hyperbolic equations without attenuation. More precisely, by leveraging the interactions induced by nonlocal attenuation, we simultaneously determine general classes of leading and lower order coefficients depending on both space and time. In addition, we formulate our results using data collected on disjoint sets, thereby extending the existing theory and introducing the notion of data on sets disjoint in both space and time for hyperbolic equations. Our approach builds on key features of nonlocal operators, such as memory effects and the unique continuation principle, while integrating the interplay between local and nonlocal operators along with structural properties of hyperbolic equations.

We determine generic completion-fiber dimensions for unit-norm tight frames when each partially observed column has exactly two missing coordinates. We work on the complex algebraic variety defined by the real frame equations. The missing coordinate pairs form a labelled multigraph. For every $d\ge4$ and every frame length $R\ge N_d:=\binom{d+1}{2}-1$, we prove that the missing-pair Jacobian matroid is the Rado matroid of an explicit subspace arrangement. Its rank is therefore given by a graph-theoretic minimum formula and is independent of $R$; the complementary rank defect is the generic fiber dimension. The main step is a realization theorem at length $N_d$. It combines a transversal–graphic matroid partition with a zero-coordinate moment submersion and a compatible decomposition of a generic residual matrix. Basis extension and stability under fully observed columns then yield the rank formula at every larger length. We also give an explicit defect formula in dimension four and a local real counterpart on the smooth real frame locus. A common-rotation obstruction bounds the smallest uniform stabilization threshold from below by $2d-1$.

The local linear minimization oracle (Ferris and Zavriev, 1996; arXiv:2605.08850), or Local LMO, solves constrained convex problems without having to compute a projection: it minimizes a linear model over the intersection of the feasible set with a ball around the current iterate. We show that, whenever the ball radius does not exceed the Polyak radius, the Local LMO step is the Euclidean projection of the current iterate onto the intersection of the feasible set with a half-space that separates the iterate from the solution set; this projection is nonetheless computable by a linear oracle alone. We demonstrate that Local LMO belongs to a broader family of projection methods which may be indexed by the depth of the localizing half-space. For objectives with $\vartheta$-Hölder continuous gradient, every method from this family whose half-space lies sufficiently deep drives the best of its first $K$ iterates to optimality at the universal rate $\mathcal{O}(K^{-(1+\vartheta)/2})$, matching the non-accelerated universal gradient method of Nesterov (2015). Run at the Polyak radius, Local LMO attains the same rate when the constrained optima are also unconstrained ($\|\nabla f(x_\star)\| = 0$), and the rate $\mathcal{O}(K^{-1/(2-\vartheta)})$ otherwise.

We consider the focusing, $L^2$-supercritical and $\dot{H}^2$-subcritical nonlinear fourth-order Schrödinger equation. The scattering of radially symmetric solutions below the ground state was proved by Guo [Comm. Partial Differential Equations (2016)] and Dinh [Nonlinearity (2021)]. In this paper, we extend the scattering results to group-invariant solutions. In Komada–Masaki [Nonlinearity (2024)], the scattering of group-invariant solutions below the ground state was proved under a certain hypothesis. To remove the hypothesis, we establish the non-optimal scattering result for general solutions, where the threshold of action is less than certain fraction of the action of the ground state. This result is analogous to that in Pausader–Shao [J. Hyperbolic Differ. Equ. (2010)] for the $L^2$-critical nonlinear fourth-order Schrödinger equation.

Deterministic neural networks and neural operators provide point predictions with no intrinsic measure of reliability. Yet, predictive uncertainty may stem from irreducible outcome variability, finite data, or limitations of the chosen model class. Monte Carlo dropout offers a computationally convenient way to construct a predictive distribution through stochastic feature masking, without training multiple independent networks or explicitly inferring a posterior over model parameters. However, its perturbation law is largely prescribed a priori and typically factorised across latent coordinates. We introduce Optimal Transport Dropout (OTD), which instead learns the predictive mapping and the law of its latent perturbations jointly. Starting from a simple independent reference distribution, OTD transports latent perturbations through a learnable flow and propagates them through the predictive neural network, thereby inducing a structured predictive law. Training uses the strictly proper Energy Score, while a kinetic-action term geometrically regularises the transport. Synthetic benchmarks show that OTD captures multimodal predictive distributions, generates meaningful dispersion when the model is misspecified, and exhibits contracting dispersion as more training data or greater model capacity are provided. For a field-valued partial differential equation surrogate, predictive dispersion strongly aligns with the spatial pattern of prediction errors. On this task, compared with Monte Carlo dropout, OTD yields more accurate predictions and better-calibrated, substantially narrower intervals. On real-world regression benchmarks, it further shows competitive accuracy and better probabilistic predictions compared to established baselines. OTD therefore offers a way to learn structured predictive uncertainty without explicit posterior inference or ensembles of independently trained predictors.

A long-standing conjecture in non-Kähler geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature should be Kähler when the constant is nonzero and Chern flat when the constant is zero. In this article, we study the corresponding problem for a Hermitian connection $\nabla$ with $\nabla T=0$. We first show that if the $\nabla$-holomorphic sectional curvature is constant, then the $(1,1)$-part of the curvature is $\nabla$-parallel. Our main result states that a nonzero constant forces $T=0$; consequently, the metric is Kähler and locally a complex space form. The proof is pointwise and requires neither compactness nor completeness. We also obtain applications to the Chern and Bismut connections.

A special semimartingale admits a unique decomposition $X=X_0+M+A$ into a local martingale $M$ and a predictable finite-variation part $A$. We consider the identification of $A$ when $X$ is observed only through repeated cross-sections. The estimand is then the projection of the sampled predictable compensator onto the observable feature filtration, namely the current state together with whatever randomness is shared across the population, so that at a fixed diffusion coefficient the marginal flow identifies the drift only up to a Markovian projection. If the drift is an affine functional of an observed lag window, the joint problem is a convex quadratic programme whose solution is the pseudo-panel regression of econometrics. Our principal concern is the case, which we believe not to have been treated before, in which the drift is the output of a hidden linear dynamical system whose dynamics are themselves to be identified from the marginals. The joint problem is then a bilinear quadratically constrained programme, which we solve to certified global optimality by spatial branch and bound; with unpenalised state disturbances and a drift basis growing with the grid it is NP-hard already in latent dimension one, by reduction from $\ell^1$ rank-one matrix approximation, whereas the complexity of the deterministic system at fixed latent dimension remains open. A block-coordinate decomposition offers a cheaper alternative. For the estimator itself, we obtain rates at a fixed mesh, separated into Monte-Carlo, estimation and grid contributions.

The deployment of unmanned aerial vehicle (UAV) systems relies on high-performance yet lightweight wireless links between UAVs and ground stations (GSs). This paper presents a geometry-aware multi-UAV in-band full-duplex (MU-IBFD) communication system that uses high-gain directional antennas and separated uplink/downlink channels to convert self-interference into controllable co-channel interference (CCI) between UAVs, thereby avoiding energy-intensive self-interference cancelers on UAVs. We also derive a geometry-aware CCI model and define a reliable operating region (ROR) in the 3D airspace, within which the SINR requirement is satisfied. A prototype consisting of two UAVs and a GS is developed, and field trials are conducted. The measured CCI as a function of UAV positions agrees well with the theoretically predicted non-ROR region, and the downlink capacity significantly exceeds that of a conventional time-division duplex (TDD) with omni-directional scheme and higher transmit power and approaches that of ideal IBFD in most of the airspace. A proof-of-concept 4K/60p video transmission further demonstrates the practical potential of the proposed MU-IBFD system.

We establish new bounds for incidences between a point set P and a family L of n-flats in PG(n+d,q). For fixed dimensions, our bound on the incidence discrepancy has an explicit piecewise-linear exponent in $\log_q|L|$, improving the classical estimate of Haemers and the Kong-Tamo bound in specified ranges of the number of flats. Matching constructions establish sharpness up to constant factors in several parameter ranges. The proof is combinatorial and avoids spectral and Fourier analytic methods. As applications, we obtain improved estimates for rich flats and exceptional orthogonal projections, together with stronger lower bounds for Furstenberg sets in certain ranges where the fraction of prescribed directions is small.

Criticality in the Ising model is conventionally generated by Hamiltonian dynamics and controlled by temperature. Here we show that critical like behavior can emerge in an Ising system whose dynamics is completely independent of the Ising Hamiltonian and contains no temperature like parameter. We introduce an adaptive cluster dynamics in which spin connectivity is controlled by a local quiet time variable, the time elapsed since a spin was last updated. The Ising Hamiltonian enters only as an observable characterizing the resulting configurations. As the parameter $\alpha$, controlling the connectivity strength, is varied, the system undergoes spontaneous symmetry breaking accompanied by strong collective fluctuations. The cluster size distribution develops an extended power law regime, terminated by a small finite size hump at the largest scales, while the Ising energy exhibits a singular response to the variation of $\alpha$. These results show that memory dependent connectivity alone can generate collective critical behavior, revealing a route to nonequilibrium criticality without energy based dynamics.

We determine the exact worst-case function-value suboptimality after $N$ first-order oracle calls on $L$-smooth, $\mu$-strongly convex functions under a nonnegative weighted combination of the initial squared distance, function-value suboptimality, and squared gradient norm. Excluding the case where no strict improvement in function-value suboptimality can be guaranteed, the exact deterministic minimax risk in dimension $d\geq2N+1$ is characterized by the unique solution of a single scalar equation involving an $N$-step recurrence. The proof constructs an explicit hard instance for the lower bound and a constant-memory first-order method, ITEM-wf (previously called ITEM-w), whose worst-case performance matches this lower bound. On the standard initial conditions, the result recovers the exact convex bound attained by the Optimized Gradient Method of Kim and Fessler, and establishes the minimax optimality of the recently introduced ITEM-f method of Kim, Ryu, and Das Gupta for the initial function-value condition.

We establish a local well-posedness theory for the $n-$dimensional Boussinesq equation with a linear restoring force in a modified Sobolev spaces $H_\omega^s$ and Bourgain-type spaces $X_\omega^{s,b}$, which are adapted to its modified dispersion relation. The restoring force introduces a new phase function whose behavior in the low-frequencies is different from ones of the classical Boussinesq equation, that in turn, affects the associated energy functional. We prove bilinear and trilinear estimates for quadratic and cubic nonlinearities, respectively, which in turn imply local well-posedness under suitable conditions on $s$ and $\omega$. To our knowledge, this is the first well-posedness result for this model.

We prove that the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ is classically free at the boundary admissible level, which confirms a conjecture of Andrews, van Ekeren, and Heluani.

An outerplanar graph is maximal if no edge can be added without losing outerplanarity. Lin and Ning determined the outerplanar graph with the largest spectral radius, and the maximizer is a maximal outerplanar graph. We determine the minimizer. In this paper, we prove that every $n$-vertex maximal outerplanar graph $G$ satisfies $\rho(G)\ge\rho(F_n)$, where $F_n$ is the zig-zag triangulation of the $n$-gon, that is, the square of the path on $n$ vertices, with equality if and only if $G=F_n$. The proof uses three local operations on maximal outerplanar graphs, each of which strictly decreases the spectral radius: the first reverses the way a piece is attached along a chord, and the second and third move a piece from one vertex to its twin across a chord when the twin carries nothing or a single ear, respectively. A graph at which no operation applies is $F_n$, or has spectral radius greater than $4$, or consists of a central triangle with three zig-zag blades of at least three triangles each and has at most $15$ vertices; in the last case it contains one of two explicit graphs on $12$ vertices whose spectral radius exceeds that of $F_{15}$. Since $\rho(F_n)<4$ for all $n$, this completes the proof. The numerical inequalities used along the way are certified by explicit integer vectors with small entries.

We give a direct, self-contained construction of the three sporadic Fischer groups $\mathrm{Fi}_{24}'$, $\mathrm{Fi}_{23}$, and $\mathrm{Fi}_{22}$ from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group $\mathrm{Fi}_{24}$. Its commutator subgroup gives $\mathrm{Fi}_{24}'$, while $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and $M_{24}$. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.

We develop a discrete Morse-Bott theory for Lefschetz complexes with real-valued incidence functions and finitely many cells in each dimension. Our main result is a reduction procedure, based on four elementary operations called Moves, for computing the remainder series $R_t$ associated with the Morse-Bott inequality. These Moves preserve $R_t$, and the procedure recursively constructs, using finitely many Moves in each dimension, a disjoint union of two-cell elementary blocks with vanishing Poincaré series. The resulting decomposition computes the coefficients of $R_t$ by counting the blocks in the corresponding dimensions. The same procedure also computes the Betti numbers of the original complex by counting the cells removed as isolated cells. As consequences, we obtain the nonnegativity of the coefficients of $R_t$ and the Morse-Bott inequality for Lefschetz complexes. The method applies in particular to CW complexes, where the nonnegativity is obtained by identifying each coefficient of $R_t$ with the number of elementary blocks in the corresponding dimension.

We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$ bounds the description complexity, $p$ is the characteristic of $\mathbb{F}$, and $1/p=0$ in characteristic zero. We also prove this bound for incidences between points and distinct geometrically irreducible components of a two-parameter polynomial family, allowing singular and nonreduced members. The proof extends Lewko's interpolation and contact-multiplicity method from lines to algebraic families. Applications include rich components, polynomial values on difference sets, and polynomial expansion.