Lines in supersingular quartics

Lines in Supersingular Quartics

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines.

The shape of spherical quartics

We discuss the problem of interpolating C Hermite data on the sphere (two points with associated first derivative vectors) by spherical rational curves. With the help of the generalized stereographic projection (Dietz et al., 1993), we construct a two– parameter family of spherical quartics solving this problem. We study the shape of these solutions and derive criteria which guarantee solutions...

Quantifier Elimination for Quartics

Concerning quartics, two particular quantifier elimination (QE) problems of historical interests and practical values are studied. We solve the problems by the theory of complete discrimination systems and negative root discriminant sequences for polynomials that provide a method for real (positive/negative) and complex root classification for polynomials. The equivalent quantifier-free formula...

Rational Points on Quartics

Supersingular Curves in Cryptography

Frey and Rück gave a method to transform the discrete logarithm problem in the divisor class group of a curve over Fq into a discrete logarithm problem in some finite field extension Fqk . The discrete logarithm problem can therefore be solved using index calculus algorithms as long as k is small. In the elliptic curve case it was shown by Menezes, Okamoto and Vanstone that for supersingular cu...