Non-vanishing forms in projective space over finite fields

Homology of Projective Space over Finite Fields

Extending Self-maps to Projective Space over Finite Fields

Using the closed point sieve, we extend to finite fields the following theorem proved by A. Bhatnagar and L. Szpiro over infinite fields: if X is a closed subscheme of P over a field, and φ : X → X satisfies φOX(1) ' OX(d) for some d ≥ 2, then there exists r ≥ 1 such that φ extends to a morphism P → P.

Pairs of Quadratic Forms over Finite Fields

Let Fq be a finite field with q elements and let X be a set of matrices over Fq. The main results of this paper are explicit expressions for the number of pairs (A,B) of matrices in X such that A has rank r, B has rank s, and A + B has rank k in the cases that (i) X is the set of alternating matrices over Fq and (ii) X is the set of symmetric matrices over Fq for odd q. Our motivation to study ...

Pencils of quadratic forms over finite fields

A formula for the number of common zeros of a non-degenerate pencil of quadratic forms is given. This is applied to pencils which count binary strings with an even number of 1’s prescribed distances apart.

Computing in Picard groups of projective curves over finite fields

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that various desirable operations can be performed efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite mor...