Diagonal equations of different degrees over p-adic fields

“…Our next lemma is a version of Hensel's Lemma, which allows us to lift a nonsingular solution of a system of congruences to a p-adic solution. This is Lemma 4 of [7]. LEMMA 3.2.…”

Section: Preliminary Lemmatamentioning

confidence: 81%

“…We prove this theorem by a method similar to the one used in [7]. First we apply a normalization process which shows that we need only consider systems that have certain desirable properties.…”

mentioning

confidence: 99%

On systems of diagonal forms

Knapp¹

2007

J. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

In this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field Q p provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in Q p . In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.

show abstract

“…In most cases, the number of variables required to establish local solubility in the current state of technology (see for example the work of Knapp [13]) is larger than what is needed to establish a local-global principle via the circle method, so we focus our attention on the latter problem. We aim for two types of Hasse principles, one for existence of solutions and one for asymptotic formulae.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Additive Equations: Repeated and Differing Degrees

Brandes

Parsell

2017

Can. j. math.

View full text Add to dashboard Cite

Abstract. We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulae, for systems of additive equations containing forms of di ering degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning-HeathBrown, which give weak results when specialized to the diagonal situation, this is the rst result on such "hybrid" systems. We also obtain specialised results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.

show abstract

Diagonal equations of different degrees over p-adic fields

Cited by 7 publications

References 8 publications

On systems of diagonal forms II

On systems of diagonal forms II

On systems of diagonal forms

Simultaneous Additive Equations: Repeated and Differing Degrees

Contact Info

Product

Resources

About