Characterizes values at infinity for real polynomial maps with 2D fibers.
problem Understanding atypical values at infinity for real polynomial maps.
method Characterization using indices of gradient vector fields on spheres.
result Analogous to two-variable case, but for maps with 2D fibers.
Analytic functions on specific domains are characterized by their smoothness and composites with polynomial curves.
problem Characterizing real analytic functions on closed subanalytic domains.
method Analyzing functions defined on closed uniformly polynomially cuspidal sets in Rn using composites with polynomial curves. result Conditions for a function to be real analytic are effectively related to the regularity of the boundary of the domain.
Study subharmonic functions in strongly symmetric Riemannian manifolds, proving polynomial growth.
problem Properties of subharmonic functions in Riemannian manifolds with a pole.
method Introduced polynomial growth of subharmonic functions and proved their properties.
result Proved polynomial growth of degree 1 for non-negative subharmonic functions.
The paper determines the bifurcation set of a real polynomial function of two variables using Newton polygons.
problem Determining the bifurcation set of a real polynomial function of two variables.
method Using toric compactification and toric modifications to count singular phenomena at infinity.
result An upper bound of the number of elements in the bifurcation set is given in terms of its Newton polygon.
Origami structures are enumerated and shown to be quantum modular.
problem Counting and understanding origami structures with real structures.
method Using combinatorics of zonal polynomials and Schur polynomials, and relating to quantum modular forms and double Hurwitz numbers.
result The generating functions of certain origami structures are quantum modular forms.
This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.
problem Determining the RLCT of sum-of-products polynomials through blow-up.
method Investigates a specific blow-up algorithm for sop polynomials to resolve their singularities.
result It is possible to resolve the singularities of sop polynomials using a specific blow-up algorithm.
The paper proves deep neural networks with analytic activation can approximate any function.
problem Approximating functions with neural networks using analytic activation functions.
method Elementary proofs for real and complex networks, Stone-Weierstrass theorem, Mergelyan's theorem.
result Closure of neural network classes equals space of polynomials for analytic activation.
Study introduces a new method for multiple parameter regularization in polynomial functional regression.
problem Handling varying regularization parameters in polynomial functional regression.
method Developed a theoretically grounded algorithm for multiple parameter regularization and model aggregation.
result Promising results from evaluations on synthetic and real-world data.
Develops methods to calculate global index of real polynomials.
problem Calculating the global index of real polynomials.
method Two methods: via atypical fibres and Milnor arcs clusters.
result Derives upper bounds for the global index, refining Durfee's degree-based bound.
Slice-polynomial functions help compute twistor discriminant loci of cubic scrolls.
problem Computing the twistor discriminant locus of cubic scrolls in CP3. method Introduced slice-polynomial functions and their companions/extensions, used twistor theory.
result Generically constant cardinality of pre-images for slice-polynomial functions.
Proof confirms conjecture for certain braids and their closures.
problem Conjecture about real algebraic links and fibered links.
method Analyzes T-homogeneous and related braids, proving conjecture for their closures.
result Conjecture confirmed for closures of T-homogeneous braids.
Proves near-tight concentration for polynomial functions of high-temperature Ising models.
problem Understanding interactions in high-dimensional data like social networks.
method Proves concentration of measure for polynomial functions of the Ising model.
result Polynomial functions of high-temperature Ising models exhibit exponential tails with optimal radius.
New method approximates partition function of graphical models using gauge functions and polynomials.
problem Computing the partition function of graphical models is computationally challenging.
method Combines gauge function technique with real stable polynomials to approximate partition function.
result Belief Propagation estimations in the sequence do not decrease and low-bound the partition function.
New method measures entanglement of open and closed curves.
problem Measuring entanglement of curves in 3-space.
method Defining bracket polynomial and Jones polynomial for open and closed curves.
result Jones polynomial applies to both open and closed curves, with continuity properties.
Paper solves a central question about nonnegative polynomials related to isoparametric polynomials.
problem Whether a given nonnegative polynomial is a sum of squares of polynomials.
method Solves the problem completely for nonnegative polynomials associated with isoparametric polynomials.
result The paper provides a complete solution for the specific case of isoparametric polynomials.
The abstract proves the non-existence of certain real algebraic surfaces.
problem The existence of real polynomial functions with specific properties.
method Algebraic solution to a problem proposed by D. A. Panov.
result There does not exist a real polynomial function with the specified properties.
New analysis shows neural networks and low-degree polynomials perform well on sparse latent structure problems.
problem Understanding the performance of neural networks and polynomial approximators on real-world sparse latent structure problems.
method Analysis of neural networks and polynomial kernels of bounded degree on a simple, natural inference problem with sparse latent structure.
result Almost-tight bounds on the performance of neural networks and low-degree polynomials for the problem, showing qualitative differences from worst-case settings.
Bayesian optimisation for expensive experiments with shape prior.
problem Expensive experiments with time-varying control variables.
method Developed a novel Bayesian optimisation framework using Bernstein polynomial basis and dynamic polynomial degree adjustment.
result Demonstrated effectiveness on polymer fibre design and learning rate optimisation.
New method realizes planar graphs as Reeb graphs of algebraic functions.
problem Realizing planar graphs as Reeb graphs of algebraic functions.
method Generic embedding and elementary procedures.
result Generically embedded planar graphs are homeomorphic to Reeb graphs of algebraic functions.
Algorithm learns neural networks with two layers in polynomial time.
problem Learning neural networks with two nonlinear layers without assumptions.
method Isotonic regression combined with kernel methods.
result First provably efficient algorithm for two-layer neural networks.
Extends Khimshiashvili's degree formula to non-isolated singularities.
problem Finding topological properties of non-isolated real singularities.
method Generalizes Khimshiashvili's topological degree formula to non-isolated singularities of real function germs.
result Algebraic formula for the Euler characteristic of fibres of real weighted-homogeneous polynomials.
The study optimizes polynomial regression for learning under Gaussian distributions.
problem Agnostic learning of Boolean and real-valued functions under Gaussian distributions.
method LP duality and polynomial degree analysis for L1-regression. result Optimal SQ lower bounds for various function classes.
The paper proves real-analyticity of superintegrable metrics and solves two conjectures.
problem Proving real-analyticity of superintegrable metrics and solving conjectures.
method Analyzing Poisson brackets and constructing new superintegrable systems.
result Proves real-analyticity of superintegrable metrics and solves two conjectures.
Study on singularities of specific polynomial functions.
problem Characterizing the topology of singularities of mixed functions.
method Introduced inner non-degenerate mixed functions and used Newton boundary to characterize links.
result Links of singularities can be completely characterized under certain conditions.
New insights into belief propagation and Bethe approximation for factor graphs.
problem Understanding the correctness and efficiency of belief propagation and its relation to partition functions.
method Viewing factor graphs through the lens of polynomials and reformulating Bethe approximation as a polynomial optimization problem.
result For bipartite normal factor graphs, the Bethe approximation is a lower bound to the partition function under certain analytic conditions.
Research on mixed polynomials, extending non-degeneracy concepts to complex variables.
problem Extending non-degeneracy concepts to mixed polynomials in complex variables.
method Generalization of Mondal's partial non-degeneracy to mixed polynomials, introducing new concepts and proving properties.
result Strong partial non-degeneracy implies isolated singularities, and mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition.
As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential forms which behave polynomially near infinity. Such polynomial differential forms naturally appear on manifolds with a c…
The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function. In particular, the nonzero level sets of a homogeneous polynomial are proper…
PolyGAN uses high-order polynomials to generate data without activation functions.
problem Learning generative models for high-dimensional distributions.
method PolyGAN models the generator as a high-order polynomial represented by high-order tensors, using tensor decompositions to reduce parameters.
result PolyGAN can approximate data distributions without activation functions.
New method estimates density functionals using polynomial basis without full distribution knowledge.
problem Estimating quantities like information divergence functions requires complete distribution knowledge and integration.
method Introduces data-driven basis functions and develops methods for basis expansions of functionals of two distributions.
result Approximates functions of distributions as closely as desired using the new basis set.
Proves non-properness set of 3D polynomial homeos can't be a line.
problem Non-properness set of 3D polynomial homeomorphisms.
method Proof by contradiction, using topological properties.
result Non-properness set cannot be homeomorphic to the real line.
New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.
problem Constructing real algebraic maps with specific properties and compositions.
method Explicit construction of real algebraic hypersurfaces and maps with prescribed images and compositions.
result Explicit families of functions represented as compositions of constructed maps with canonical projections.
Spaces of polynomials are shown to be Euclidean balls.
problem Understanding the geometry of Lorentzian and real stable polynomials.
method Refined connection between symmetric exclusion process and polynomial geometry.
result Spaces of Lorentzian and real stable polynomials are homeomorphic to closed Euclidean balls.
The paper studies conditions for graphs connecting level sets of harmonic polynomials.
problem Conditions for graphs connecting level sets of harmonic polynomials.
method Algebraic properties and Kempf-Ness functional construction.
result Stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation.
Constructs real algebraic functions with specified preimages.
problem Reconstructing smooth functions with prescribed preimages.
method Using real algebraic functions and techniques from singularity theory and differential topology.
result Constructs examples of real algebraic functions with specified preimages.
In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and…
Formula for umbilic points on polynomial surfaces, proving their isolated nature and topological type.
problem Understanding the global behavior of fields of principal directions on polynomial surfaces.
method Poincaré-Hopf type formula and projective extension analysis.
result Every umbilic point at infinity has index 1/2 and topological type a Lemon.
The Jacobian conjecture is simplified using polynomial mappings.
problem Simplifying the Jacobian conjecture over the real field.
method Using polynomial mappings to restrict transitions on manifolds.
result An equivalent statement of the Jacobian conjecture.
Geodesics in jet space are constructed from polynomials, with some yielding globally minimizing paths.
problem Characterize geodesics in jet space and identify those that are globally minimizing.
method Sub-Riemannian geometry, Hamilton-Jacobi equations, and analysis of period degenerations.
result Some polynomials yield globally minimizing geodesics, with conjectures on the independence of cut time.
The paper encodes local shapes of polynomial curves using permutations.
problem Measuring non-convexity of real algebraic plane curves.
method Generic projections avoiding specific tangencies.
result Local shapes of curves can be encoded in alternating permutations.
New knot models analyze local entanglement for robust curve analysis.
problem Lack of local structural information in classical knot theory.
method Proposed multiscale and persistent Jones polynomials.
result Models are stable to small perturbations, robust for real-world applications.
A polynomial knot is a smooth embedding κ:ℜ→ℜn whose components are polynomials. The case n=3 is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological knot. This paper contains basic results for these knots as well as many examples.
The paper defines functions that induce bounded composition operators on RKHSs with analytic positive definite functions.
problem Characterizing functions that induce bounded composition operators on RKHSs.
method Intrinsic properties of RKHSs and asymptotic properties of orthogonal polynomials.
result Only affine transforms can induce bounded composition operators in a large class of RKHSs.
Study volume growth in Milnor fibers using real Lagrangians.
problem Volume growth in Milnor fibers of Brieskorn polynomials.
method Investigate real Lagrangians, use Smith inequality for involutions in wrapped Floer homology.
result Uniform lower bound of volume growth for a class of Brieskorn polynomials.
Polynomials' roots count tied to surface umbilics.
problem Relating roots of polynomials to umbilics on surfaces.
method Constructing a convex surface from a polynomial, determining umbilic index, and applying Hamburger's bound.
result Bounding the number of roots inside the unit circle for polynomials with self-inversive second derivatives.
Let g:X -> Y be a smooth (i.e. C^\infty differentiable) map between two smooth manifolds. In analogy with the case of complex polynomial functions, we say that y_0 in Y is a typical value of g if there exists an open neighbourhood U of y_0 in Y, such that the restriction g:g^{-1}(U) -> U is a C^\infty trivial fibration…
Quantum dilogarithms help define invariants of 3-manifolds.
problem Defining invariants of 3-manifolds using quantum dilogarithms.
method Associate quantum dilogarithms to local fields, construct TQFTs, and use partition functions.
result Quantum dilogarithms yield invariants related to A-polynomial curves. The paper analyzes the asymptotic behavior of a knot polynomial for a specific real number.
problem Understanding the asymptotic behavior of a knot polynomial for a real number.
method Examining the asymptotic behavior of the N-dimensional colored Jones polynomial evaluated at exp(ξ/N) for a real number ξ. result From the asymptotic behavior, the mSL(2;C) Chern--Simons invariant and the Reidemeister torsion twisted by the adjoint action can be extracted.