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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Motivated by Emmanuel Kowalski's exponential sums over definable sets infinite fields, we generalize Ax's theorem on pseudo-finite fields to acontinuous-logic setting allowing for an additive character. The role played byWeil's Riemann hypothesis for curves over finite fields is taken by the `Weilbound' on exponential sums. Subsequent model-theoretic developments, includingsimplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, alsogeneralize.  Analytically, we have the following consequence: consider the algebra offunctions $\Ff_p^n \to \Cc$ obtained from the additive characters and thecharacteristic functions of subvarieties by pre- or post-composing withpolynomials, applying min and sup operators to the real part, and averagingover subvarieties. Then any element of this class can be approximated,uniformly in the variables and in the prime $p$, by a polynomial expression in$\Psi_p(\xi)$ at certain algebraic functions $\xi$ of the variables, where$\Psi(n \mod p) = exp(2 \pi i n/p)$ is the standard additive character.

Ax's theorem with an additive character