The zeros of a quadratic form at square-free points

On the square root of quadratic matrices

Here we present a new approach to calculating the square root of a quadratic matrix. Actually, the purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine an explicit formula for all the square roots of $2times 2$ matrices.

Inequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$‎, ‎let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| leq k, (kleq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$alphainmathbb{C}$ with $|alpha|geq k$‎, ‎begin{align*}‎ ‎max_{|z|=...

Small Zeros of Quadratic Forms

Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ Q N an L-dimensional subspace, 1 ≤ L ≤ N . We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F ). This provides an analogue over Q of a wellknown theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bil...

Hyperinterpolation at Xu Points and Interpolation at Padua Points in the Square: Computational Aspects

Garside Structure on Monoids with Quadratic Square-free Relations

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were i...