Quartic Forms in Many Variables

Dumke, Jan H.

doi:10.48550/arxiv.1405.7064

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2015

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The second bound follows from a recent result by Dumke [7] which establishes γ * 3 (2) ≤ 131. Previously, the best known bound was a result by Dietmann and Wooley [6], who by a different approach showed that any two cubic forms, not necessarily nonsingular, have a simultaneous zero if the number of variables is at least 827.…”

Section: Introductionmentioning

confidence: 86%

A note on p -adic solubility for forms in many variables

Brandes

2015

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity of the singular series. This improves on earlier approaches due to Davenport, Schmidt and others, and allows us to establish an asymptotic formula for the number of simultaneous zeros of non-singular pairs of cubic forms in at least 131 variables. As a by-product, we obtain a version of Hensel's Lemma for linear spaces.

show abstract

Section: Introductionmentioning

confidence: 86%