Artin's conjecture for septic and unidecic forms

$p$-adic zeros of quintic forms

“…He proved that a quintic form over Q p possesses a non-trivial zero if p ≥ 17. For septic and unidecic forms bounds q 0 (7) ≤ 883 and q 0 (11) ≤ 8053 are due to Wooley [13]. In this paper we shall establish q 0 (5) ≤ 9.…”

Simultaneous zeros of a cubic and quadratic form

“…We are now in a position to find a non-singular zero x 2 for H where x 2 = 0, which, by Lemma 5.10, implies that there is a non-singular zero of F. To do this, we employ a slicing approach, following an idea used by Wooley [27] for the case of degree 7 and 11 forms. In our context Wooley's method enables one to find a non-singular point more efficiently, than if one applied the crude point counting estimates which are currently available for hypersurfaces.…”

“…For example, following the work on cubic forms, Birch and Lewis [3] and later Laxton and Lewis [16], respectively, dealt with the cases of forms of degree 5 and later degrees 5, 7 and 11, obtaining a condition on the cardinality of the residue class field. Later work by Leep and Yeomans [18] (degree 5), Knapp [15] (degrees 7 and 11), Wooley [27] (degrees 7 and 11) and most recently Heath-Brown [14] (degree 5) have provided us with bounds for the cardinality of the residue class field that fall narrowly short of the truth.…”

Nonsingular points on hypersurfaces over $ \mathbb{F}_q $

“…A reasonable lower bound for the characteristic required is only known in a handful of cases. For example, recently Wooley [11] has shown that for d = 7 and 11 we require the size of the residue class field to respectively exceed 883 and 8053. These estimates depend on the ability to ascertain the existence of a non-singular point on the F q -varieties under consideration.…”