Related papers: On Projective Equivalence of Univariate Polynomial…

Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\ball)$, $1<p<\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces…

There are numerous combinatorial objects associated to a Grassmannian permutation $w_\lambda$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations…

We consider the problem of finding for a given $N$-tuple of polynomials (real or complex) the closest $N$-tuple that has a common divisor of degree at least $d$. Extended weighted Euclidean seminorm of the coefficients is used as a measure…

We study subvarieties of a general projective degree $d$ hypersurface $X_d\subset \mathbf P^n$. Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety…

The named space denoted by $V_{pq}^k$ consists of $L_q$ functions on $[0,1)^d$ of bounded $p$-variation of order $k\in\mathbb N$. It generalizes the classical spaces $V_p(0,1)$ ($=V_{p\infty}^1$) and $BV([0,1)^d)$ ($V_{1q}^1$ where…

Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper…

In this paper we introduce the systematic study of invariant functions and equivariant mappings defined on Minkowski space under the action of the Lorentz group. We adapt some known results from the orthogonal group acting on the Euclidean…

The aim of this paper is twofold. The first is to give a quantitative version of Schmidt's subspace theorem for arbitrary families of higher degree polynomials. The second is to give a generalization of the subspace theorem for arbitrary…

Let F : W --> V be a dominant rational map between quasi-projective varieties of the same dimension. We give two proofs that h_V(F(P)) >> h_W(P) for all points P in a nonempty Zariski open subset of W. For dominant rational maps F : P^n -->…

Working in a polynomial ring $S=\mathbf{k}[x_1,\ldots,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\geq r, d)}$…

We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…

The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n-d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of…

Recent work has shown the surprising power of low-degree sandwiching polynomial approximators in the context of challenging learning settings such as learning with distribution shift, testable learning, and learning with contamination. A…

Let $F$ be a finite field with the characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis for $\mathbb{Z}_{2}$-graded…

For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the intersection E-polynomials and the intersection Poincar\'e polynomials of the $G$-character variety of a compact Riemann surface $C$ and of the moduli space of $G$-Higgs…

We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$…

On the Grassmann manifold G (m, n) of m-dimensional subspaces of an n-dimensional projective space P^n, a certain supplementary construction called the normalization is considered. By means of this normalization, one can construct the…

We prove that for any norm |*| in the d-dimensional real vector space V and for any odd n>0 there is a non-negative polynomial p(x), x in V of degree 2n such that p^{1/2n}(x) < |x| < c(n,d) p^{1/2n}(x), where c(n,d)={n+d-1 choose n}^{1/2n}.…

A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal…

The Plucker relations define a projective embedding of the Grassmann variety Gr(k,n). We give another finite set of quadratic equations which defines the same embedding, and whose elements all have rank 6. This is achieved by constructing a…