Diagonal equations over unramified extensions of Qp

Leep, David B.; Vieira, Luis Sordo

doi:10.1112/blms.12163

Cited by 4 publications

(3 citation statements)

References 13 publications

(24 reference statements)

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“…This is well known. It is also a special case of the version of Hensel's Lemma employed by Leep and Sordo Vieira [7, Theorem 2.3]. We include a proof of this lemma in the interest of completeness.…”

Section: Preliminaries On P‐adic Local Fieldsmentioning

confidence: 97%

“…We begin with a simple lemma: This lemma is essentially [7,Theorem 2.1]. We include a proof for completeness.…”

Section: Proof Of Theorem A: the Case R =mentioning

confidence: 99%

“…The bound in Theorem A has the merit of being both independent of the field

K

and essentially uniform in

d

and

R

, though better bounds can sometimes be achieved under additional restrictions on

K

d

, or

R

. For example, if

K

is unramified over

Q_{p}

and

R = 1

, then Leep and Sordo Vieira 7 have shown that

s > d^{2}

suffices, which is the bound predicted by Artin's Conjecture for homogeneous forms. Of course, Artin's Conjecture optimistically predicts that

s > R d^{2}

suffices for solvability of any system ().…”

Section: Introductionmentioning

confidence: 97%

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

Skinner

2020

Proc. London Math. Soc.

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Section: Preliminaries On P‐adic Local Fieldsmentioning

confidence: 97%

“…We begin with a simple lemma: This lemma is essentially [7,Theorem 2.1]. We include a proof for completeness.…”

Section: Proof Of Theorem A: the Case R =mentioning

confidence: 99%

“…The bound in Theorem A has the merit of being both independent of the field

K

and essentially uniform in

d

and

R

, though better bounds can sometimes be achieved under additional restrictions on

K

d

, or

R

. For example, if

K

is unramified over

Q_{p}

and

R = 1

, then Leep and Sordo Vieira 7 have shown that

s > d^{2}

suffices, which is the bound predicted by Artin's Conjecture for homogeneous forms. Of course, Artin's Conjecture optimistically predicts that

s > R d^{2}

suffices for solvability of any system ().…”

Section: Introductionmentioning

confidence: 97%

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

Skinner

2020

Proc. London Math. Soc.

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Solubility of additive forms of twice odd degree over totally ramified extensions of Q2$\mathbb {Q}_2$

Duncan

2024

Bulletin of London Math Soc

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Solubility of additive quartic forms over ramified quadratic extensions of ℚ2

Duncan

Leep

2022

Int. J. Number Theory

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In this paper, we determine the minimal number of variables [Formula: see text] which guarantees a nontrivial solution for every additive form of degree [Formula: see text] over the four ramified quadratic extensions [Formula: see text] of [Formula: see text]. In all four fields, we prove that [Formula: see text]. This is the first example of such a computation for a proper ramified extension of [Formula: see text] where the degree is a power of [Formula: see text] greater than [Formula: see text].

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Diagonal equations over unramified extensions of Qp

Abstract: If K is an unramified extension of the p‐adic field Qp where p⩾3, then we prove that any diagonal homogeneous form defined over K of degree d has a nontrivial zero in K provided that the number of variables is greater than d2.

Cited by 4 publications

References 13 publications

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

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

Solubility of additive forms of twice odd degree over totally ramified extensions of Q2$\mathbb {Q}_2$

Solubility of additive quartic forms over ramified quadratic extensions of ℚ2

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