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.
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.
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.
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.
We prove that the set of non-properness of a polynomial mapping of the three dimensional space which is a local homeomorphism cannot be homeomorphic to the real line R.
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.
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected components, and its representation should be the direct sum of three irreducible two-dim…
Continuity of roots of hyperbolic polynomials with smooth coefficients.
problem Continuity of the solution map for hyperbolic polynomials.
method Proving continuity of the solution map from hyperbolic polynomials of degree d with C^d coefficients to their increasingly ordered roots.
result Continuity of the solution map for hyperbolic polynomials with C^d coefficients.
Study connects spectral properties to frame flows on curved manifolds.
problem Spectral properties and frame flows on curved manifolds.
method Link between spectral properties, frame flows, and polynomial maps between spheres.
result Ergodicity of frame flows on low-rank bundles.
The paper proves Sard's theorem for polynomial maps in infinite dimensions.
problem The validity of Sard's theorem for polynomial maps in infinite-dimensional Banach manifolds.
method Sharp quantitative criteria for the validity of Sard's theorem.
result The paper provides criteria for the validity of Sard's theorem in infinite-dimensional Banach manifolds.
Maps and embeddings between hyperbolic spaces and their boundaries studied.
problem Understanding relations between maps and embeddings between relatively hyperbolic spaces and their boundaries.
method Establishing correspondences between quasi-isometric embeddings and quasisymmetric embeddings, using polynomial distortion.
result Characterization of hyperbolic relative groups with polynomial distortion embeddings.
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.
The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Let f be a 1-variable complex polynomial such that f has a singularity at the origin. In the present paper, we show that there exists a deformation of f which has only fold singularities and cusps as singularities of a real polynomial map from the plane to the plane. We then calculate the number of cusps of a deformati…
In this paper, we study deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and prove that the deformed polynomial maps have only indefinite fold and cusp singularities in general. We then estimate the number of cusps appearing in such…
Recent years have demonstrated that using random feature maps can significantly decrease the training and testing times of kernel-based algorithms without significantly lowering their accuracy. Regrettably, because random features are target-agnostic, typically thousands of such features are necessary to achieve accept…
We consider the moduli space of polystable L-twisted G-Higgs bundles over a compact Riemann surface X, where G is a real reductive Lie group, and L is a holomorphic line bundle over X. Evaluating the Higgs field at a basis of the ring of polynomial invariants of the isotropy representation, one defines the …
Continuity of polynomial roots shown for varying coefficients.
problem Continuity of polynomial roots under varying coefficients.
method Uniform bounds and Sobolev space analysis.
result Solution map is continuous for Cd coefficients. Given a polynomial map ψ:Sm→Rk with components of degree d, we investigate the structure of the semialgebraic set Z⊆Sm consisting of those points where ψ and its derivatives satisfy a given list of polynomial equalities and inequalities (we call such a set a "singularity"). Concerning th…
We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …
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.
Introduces quasi-holomorphic maps and their properties.
problem Understanding singularities and stratifications in non-complex manifolds.
method Pontryagin--Thom construction, cobordism groups, Thom polynomials.
result Thom polynomials determine cohomology classes of quasi-holomorphic maps.
Formula calculates homology groups of Milnor fibres for real hypersurface singularities.
problem Calculating homology groups of Milnor fibres for real hypersurface singularities.
method Established a formula for homology groups of Milnor fibres relative to their boundaries.
result Formula provides a method to compute homology groups of Milnor fibres.
We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N=TM carries a canonical structure of (affine) special Kähler manifold. This gives a…
We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements.
A new method reduces the complexity of tensor products from cubic to quadratic, improving both speed and accuracy.
problem Efficiently computing high-dimensional tensor products for polynomial kernels.
method Complex-to-Real (CtR) modification of sketches using complex random projections.
result Achieves state-of-the-art performance in accuracy and speed.
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.
Cyclotomic polynomials help classify mapping classes on surfaces.
problem Characterizing mapping classes on surfaces using cyclotomic polynomials.
method Investigating characteristic polynomials of integral symplectic matrices and using cyclotomic polynomials to classify them.
result For n≥3, the polynomial φn(x) is realized by a mapping class of algebraically finite type if and only if n has at most two distinct prime divisors. Milnor fibrations were extended by Mutsuo Oka for certain mixed polynomial. In this paper, we study singular points of differentiable maps into the 2-dimensional torus, called Milnor fibration product maps, obtained by several Milnor fibrations for mixed polynomial. We give a characterization of singular points of such…
Proves polynomial injectivity of Fubini-Study map for ample line bundles.
problem Injectivity of Fubini-Study map for ample line bundles.
method Polynomial injectivity proof with polynomial dependence on ample line bundle exponent.
result Quantitative version of injectivity proved, polynomial in ample line bundle exponent.
Relative Thom polynomials for maps around boundaries established.
problem Understanding singularities in maps around boundaries.
method Introducing and analyzing Thom polynomials relative to prescribed maps around boundaries, establishing structure theorems and correction terms.
result Unified framework for invariants of immersions and singularities of their extensions.
Abstract: Study of geometric structures on surfaces using various tools.
problem Understanding geometric structures on surfaces.
method Use of volume, contact, symplectic, complex, and almost complex structures; local rigidity results; higher-dimensional analogues; constructions with Riemann surfaces; definitions using surjective homomorphisms; models of hyperbolic plane and 3-space; conformal structures.
result Introduction of new models and constructions for hyperbolic plane and 3-space.
Smooth maps bound Betti numbers of zero sets.
problem Bounding Betti numbers of zero sets of smooth maps.
method Generalized Thom-Milnor bound to polynomial maps on nonsingular real algebraic varieties; introduced condition number for families of functions.
result Extended Thom-Milnor bounds to families of functions and semialgebraic sets.
Constructs real algebraic maps with specific geometric constraints.
problem Construct smooth functions with prescribed Reeb graphs.
method Explicitly constructs real algebraic maps whose images are domains surrounded by products of hyperbolas and affine spaces.
result New examples of real algebraic maps with specified geometric constraints.
Solves generalized twisted rabbit problems for higher degree polynomials.
problem When a quadratic polynomial is twisted by a cyclic subgroup, what polynomial is equivalent?
method Uses d2-adic expansion instead of 4-adic for higher degree polynomials. result Provides a solution that depends on the d2-adic expansion of the power of the mapping class element. We use topology of configuration spaces to give a characterization of Neuwirth--Stallings pairs (S5,K) with dimK=2. As a consequence, we construct polynomial map germs (R6,0)→(R3,0) with an isolated singularity at the origin such that their Milnor fibers are not diffeomorphic to a…
The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.
The study characterizes and constructs polynomial harmonic morphisms on spheres.
problem Characterizing and constructing polynomial harmonic morphisms on spheres.
method Characterization and construction of polynomial harmonic morphisms using eigenfamilies.
result Strong restrictions and classification of polynomial harmonic morphisms in low dimensions.
This paper characterizes stable polynomial mappings in a specific set.
problem Characterizing stable polynomial mappings in a given set.
method Analyzing polynomial mappings with specific degrees and determining topological equivalence.
result Effective determination of mappings with generic topology.
Polynomial representations found in surface braid and mapping class groups.
problem Homological representations of surface braid and mapping class groups.
method Study of homological representation functors and short exact sequences.
result Many homological representation functors are polynomial.
Proves transitivity of a specific class of quadratic polynomials.
problem Transitivity of pure Hurwitz classes of post-critically finite quadratic polynomials.
method Uses mapping classes of the sphere with finitely many marked points.
result Establishes transitivity for pure Hurwitz classes of post-critically finite quadratic polynomials.
Recent proofs of classical theorems in polynomial algebra and functional analysis are discussed, which use tools from the topology of real manifolds. Simpler proofs were discovered in the new century, of the Hilbert Nullstellensatz, and the Gelfand-Mazur Theorem. We give a related proof that an irreducible real polynom…
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
Polynomial maps are shown to be Serre fibrations under specific conditions.
problem Characterizing polynomial maps as Serre fibrations.
method Using relative homotopy groups and analyzing polynomial maps over simple arcs.
result Polynomial maps are Serre fibrations over certain simple arcs.
Automatic continuity of polynomial maps and cocycles proved.
problem Proving continuity of polynomial maps and cocycles.
method The approach involves proving continuity of polynomial maps and cocycles.
result Automatic continuity of polynomial maps and cocycles.
The paper characterizes biharmonic maps between spheres using polynomial functions.
problem Characterizing biharmonic maps between spheres using polynomial functions.
method Proved a characterization formula and constructed biharmonic maps.
result Classification of all proper biharmonic quadratic forms from spheres.
In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal s…
Researchers map the fundamental group of polynomial strata to a braid group.
problem Understanding the fundamental group of polynomial strata.
method Analyzing the logarithmic derivative of polynomials to determine the map to a braid group.
result The map from the fundamental group of a stratum to a braid group is characterized by the geometry of the translation surface structure.