English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let p be a prime and K the algebraic closure of the finite field GFepT. We will always work in characteristic p and consider P, as a scheme over GFepT. Let X be an alge-braic scheme defined over a finite field GFep,T. X eKT will denote the set of all K-points of X. For every power q of p with q d p, let X eqT denote the set of all GFeqT-points of X. Hence X eqT J X eq,T if q, q, are p-powers and q, dqdp,. X eKT is the union of all X eqT, q g 0 and q a p-power. If X is reduced, then the scheme X is uniquely deter-mined by the algebraic variety X eKT in the sense of Serre (Hilbert Nullstellensatz). If X is not a zero-dimensional scheme, then X eKT is infinite. We fix a p-power q with q d p, and we would like to see up to what order the finite set X eqT determines the infinite set X eKT. Now assume that X is projective and that it is equipped with an embedding XHP, defined over GFeqT. Let k be an integer. We say that the pair eX ; X eqTT satisfies the Finite Field Nullstellensatz of order k (or just that FFNekT is true for X and X eqT) if every homogeneous form of degree c k on P,eKT vanishing on X eqT vanishes on X eKT. Choose homogeneous coordinates z0;... ; zN on P,. The set PGeN; qT is the union of q t 1 hyperplanes; for instance take the hyperplanes z0 ? czN, c A GFeqT, and the hyperplane zN ? 0.Hence if X eKT 0 X eqT (and in particular if dimeX T > 0), then the pair eX ; X eqTT does not satisfy FFNeq t 1T. A. Blokhuis and G. E. Moor-house proved FFNeq 1T for an elliptic quadric surface, FFNeqT for a hyperbolic quadric surface and FFNeqT for a smooth quadric hypersurface of PGen; qT, q d 4 [1]. G. E. Moorhouse proved FFNeqT for Hermitian varieties, q a square [5, Theorem 4.1], and FFNeq 1T for Grassmann varieties [6, O4]. Here we consider the case of the intersection of two quadric hypersurfaces and prove the following result.

On the Finite Field Nullstellensatz for the intersection of two quadric hypersurfaces