Related papers: Stratifying Discriminant Hypersurface
A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
The Euler characteristic of a very affine variety encodes the number of critical points of the likelihood equation on this variety. In this paper, we study the Euler characteristic of the complement of a hypersurface arrangement with…
We characterize all possible relative positions between a hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. The classification is also suitable for a hyperboloid and a…
Recent progress in contrastive learning has revolutionized unsupervised representation learning. Concretely, multiple views (augmentations) from the same image are encouraged to map to the similar embeddings, while views from different…
Covariance pooling is a feature pooling method with good classification accuracy. Because covariance features consist of second-order statistics, the scale of the feature elements are varied. Therefore, normalizing covariance features using…
Hyperspectral analysis has gained popularity over recent years as a way to infer what materials are displayed on a picture whose pixels consist of a mixture of spectral signatures. Computing both signatures and mixture coefficients is known…
We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by…
In this paper, we study the persistent homology of the offset filtration of algebraic varieties. We prove the algebraicity of two quantities central to the computation of persistent homology. Moreover, we connect persistent homology and…
Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric…
We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the…
We study the link between a compact hypersurface in $\P^{n+1}$ and the set of all its tangent planes. In this context, we identify $\P^{n+1}$ to the set of linear subspaces of codimension one by orthogonal complementarity. This gives rise…
We utilise the two principles of decoupling introduced in arXiv:2407.16108 to prove the following conditional result: assuming uniform decoupling for graphs of polynomials in all dimensions with identically zero Gaussian curvature, we can…
This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spaces of the exact manifolds, and they have…
Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper…
We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.
We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous…
We present a formula for the degree of the discriminant of irreducible representations of a Lie group, in terms of the roots of the group and the highest weight of the representation. The proof uses equivariant cohomology techniques,…
An emerging class of trajectory optimization methods enforces collision avoidance by jointly optimizing the robot's configuration and a separating hyperplane. However, as linear separators only apply to convex sets, these methods require…
We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of…