Higher Gauss maps of Veronese varieties—A generalization of Boole’s formula and degree bounds for higher Gauss map images

A Note on Higher Order Gauss Maps

We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L. We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X , with k > 1, then the Gauss map of order k has finite fibers, unless X = Pn is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a com...

L_1 operator and Gauss map of quadric surfaces

The quadrics are all surfaces that can be expressed as a second degree polynomialin x, y and z. We study the Gauss map G of quadric surfaces in the 3-dimensional Euclidean space R^3 with respect to the so called L_1 operator ( Cheng-Yau operator □) acting on the smooth functions defined on the surfaces. For any smooth functions f defined on the surfaces, L_f=tr(P_1o hessf), where P_1 is t...

Higher Order variance and Gauss Jacobi Quadrature

In this report, we study in a detailed way higher order variances and quadrature Gauss Jacobi. Recall that the variance of order j measures the concentration of a probability close to j points x j,s with weight λ j,s which are determined by the parameters of the quadrature Gauss Jacobi. We shall study many example in which these measures specify adequately the distribution of probabilities. We ...

Generalization of a Quadratic Transformation Formula due to Gauss

The Gauss Higher Relative Class Number Problem

Assuming the 2-adic Iwasawa main conjecture, we find all CM fields with higher relative class number at most 16: there are at least 31 and at most 34 such fields, and exactly one is not abelian. The problem of determining all imaginary quadratic fields K = Q( √ d) of small class number h(K) was first posed in Article 303 of Gauss’ Disquisitiones Arithmeticae. It would take almost 150 years of w...