Artin's conjecture and systems of diagonal equations

Abstract

Abstract. We show that Artin's conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r 2.

Solvability of systems of diagonal equations over p‐adic local fields

“…Of course, Artin's Conjecture optimistically predicts that suffices for solvability of any system (). Determining the truth of this prediction for systems over a general ‐adic field (including is still an interesting and open problem (but see also 14 for some negative results).…”

On Artin’s conjecture: Linear slices of diagonal hypersurfaces

“…For larger R little is known (see [4,10,20]). It is rather remarkable that Wooley [28] very recently found examples with R = 2 where the conjecture fails, though such failures have been familiar for large R (see [19]). From the perspective taken here, our theorem adds to the small stock of examples where a conjecture of Artin's type has been verified for a class of forms of even degree.…”

Two Cases of Artin's Conjecture

“…, k R ) for all but a finite set of primes. But in general, it follows from a result by Lewis and Montgomery [31,Theorem 2] that this conjecture is not true and, furthermore, Wooley [42] proved that even the case R = 2 does not hold for all tuples (k 1 , k 2 ). However, there are cases in which it does hold.…”