arXiv research

The paper explores how polynomial roots and operator eigenvalues change with parameters.

A new method builds sparse polynomial chaos expansions for models with dependent inputs.

New algorithm speeds up polynomial kernel approximations.

Solves generalized twisted rabbit problems for higher degree polynomials.

New SDEs from affine and polynomial perspectives for path-dependent processes.

We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.

Link signature limit depends on linking matrix under specific polynomial condition.

Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using …

New invariant for tied links connects states without resolution dependence.

Study geometric bases for A-polynomials in SU(3) using arcade formalism.

The study computes trace fields and minimal polynomials for specific knots and links.

We show that the zeroes of the Alexander polynomial of a Lorenz knot all lie in some annulus whose width depends explicitly on the genus and the braid index of the considered knot.

Polynomial bound on surfaces in hyperbolic 3-manifolds.

We explain an algorithm for finding a boundary link Seifert matrix for a given Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.

We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to give a state sum formula for the twisted Alexander polynomial of a knot dependin…

Study on Monge-Ampère equations with polynomial growth rates.

Machine learning improves Buchberger's algorithm for polynomial systems.

We prove a new lower bound for the dilatation of an arbitrary pseudo-Anosov map on a surface of genus g with n punctures. Our bound improves the former super-exponential dependence on the genus by a polynomial dependence.

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irr…

Researchers compute Khovanov polynomials for satellite knots.

We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerbo…

We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in…

Solves infinite family of cubic polynomial problems.

Robustly estimates multivariate polynomials in noisy data.

Polynomial convergence proved for SGM, improving over previous methods.

We study the dependence of solutions of equations of the form $a_0 + a_1 z^{\ell_1} + ... + a_m z^{\ell_m} = 0$, on the exponents $\ell_1, ..., \ell_m$. We apply our results to equations that appear in graph theory, the theory of 3-manifolds fibering over the circle, and the theory of free-by-cyclic groups. In particul…

Skew parallelogram nets factorize, encompassing discrete differential geometry.

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accura…

Bayesian neural networks learn efficiently at infinite width, matching polynomial-width performance.

The colored HOMLFY polynomial is an important knot invariant depending on two variables $a$ and $q$. We give bounds on the degree in both $a$ and $q$ generalizing Morton's bounds \cite{Mo86} for the ordinary HOMFLY polynomial. Our bounds suggest that the degree detects certain incompressible surfaces in the knot comple…

We investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3. These solutions are parametrized by complex matrix-valued polynomials called potentials. On the space of these potentials there act two commuting flows. The orbits of these flows are called Polynomial Killing fields and are double peri…

The paper categorifies matroid characteristic polynomials using cohomology.

Algorithm creates polynomials for knotted surfaces, with bounds on degree.

Derives a series expansion for Asian option pricing with polynomial jump-diffusion moments.

Study on future-dependent value functions for off-policy evaluation in complex environments.

Classifies small links in an unmarked solid torus.

We give a global version of Le-Ramanujam mu-constant theorem for polynomials. Let f_t, (t in [0,1]), be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number, the Mi…

This work reconstructs knot invariants from Alexander polynomials, proving consistency with known theorems.

Paper reduces sample complexity for bilinear systems identification to nearly constant.

Analyzes the differential expansion of knot polynomials, focusing on its applicability and modifications.

The paper shows how neural networks can approximate PDEs with polynomial scaling in dimension.

In the setting of polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for p…

New invariants defined for knots and links using quandle representations.

The paper addresses online prediction in marginally stable systems with bounded perturbations.

New framework allows reinforcement learning with polynomial sample complexity.

We study relations between the Alexander-Conway polynomial $\nabla_L$ and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of $\nabla_L$ of an m-component link L all of whose Milnor numbers $μ_{i_1... i_p}$ van…

Exact causal network discovery is polynomial for sparse networks.