Forms representing forms and linear spaces on hypersurfaces

Brandes, Julia

doi:10.1112/plms/pdt044

Cited by 20 publications

(60 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Birch's work [3] is generalised to systems of forms with differing degrees by Browning and Heath-Brown [7] over Q and by Frei and Madritsch [16] over number fields. It is extended to linear spaces of solutions by Brandes [5,6]. Versions of the result for function fields are due to Lee [22] and to Browning and Vishe [9].…”

Section: Related Workmentioning

confidence: 99%

Quadratic forms and systems of forms in many variables

Myerson

2018

Invent. math.

View full text Add to dashboard Cite

Let F 1 , . . . , F R be quadratic forms with integer coefficients in n variables. When n ≥ 9R and the variety V (F 1 , . . . , F R ) is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for V (F 1 , . . . , F R ). Previous work in this direction required n to grow at least quadratically with R. We give a similar result for R forms of degree d, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as n > d2 d R. In the case that d ≥ 3, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients. Mathematics Subject Classification

show abstract

Section: Related Workmentioning

confidence: 99%

Quadratic forms and systems of forms in many variables

Myerson

2018

Invent. math.

View full text Add to dashboard Cite

show abstract

“…More general versions of each of the cases of Theorem 1.1 are available below (see Theorems 4.1, 7.1) that somewhat relax the requirement that F should be smooth. The glaring omission here is of course the case d = 2m; while the analytical aspects of the treatment of this case are in fact more conventional than in the situation when d = 2m and largely follow the arguments of [1,2], the geometry creates additional difficulties when the singularities of F interfere with the discriminant equation. We plan to resolve this issue in future work.…”

Section: Introductionmentioning

confidence: 99%

“…Fortunately, this can be avoided by pruning instead a different set of major arcs that can be defined for any positive θ 1. We record here Lemma 3.5 of [2], which serves as starting point for our first pruning step. …”

Section: The Minor Arcs In the Case D > 2mmentioning

confidence: 99%

“…We plan to resolve this issue in future work. We will prove Theorem 1.1 by the circle method via a combination of the ideas presented in our own work [2] and recent work by Browning and Heath-Brown [5]. Observe that the latter can be applied directly to the simultaneous Eqs.…”

Section: Introductionmentioning

confidence: 99%

“…. , x s ] be a form of degree d. In previous work [2,4] we investigated the number of rational linear spaces of dimension m contained in the hypersurface given by . for some non-negative constants χ ∞ and χ p characterising the density of solutions over the local fields R and Q p , respectively, provided that…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the number of linear spaces on hypersurfaces with a prescribed discriminant

Brandes

2017

Math. Z.

Self Cite

View full text Add to dashboard Cite

For a given form F ∈ Z[x 1 , . . . , x s ] we apply the circle method in order to give an asymptotic estimate of the number of m-tuples x 1 , . . . , x m spanning a linear space on the hypersurface F(X ) = 0 with the property that det ((x 1 , . . . , x m ) t (x 1 , . . . , x m )) = b. This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.

show abstract

A Variant of Weyl’s Inequality for Systems of Forms and Applications

Schindler¹

2015

Fields Institute Communications

View full text Add to dashboard Cite

Abstract. We give a variant of Weyl's inequality for systems of forms together with applications. First we use this to give a different formulation of a theorem of B. J. Birch on forms in many variables. More precisely, we show that the dimension of the locus V * introduced in this work can be replaced by the maximal dimension of the singular loci of forms in the linear system of the given forms. In some cases this improves on the aforementioned theorem of Birch.We say that a system of forms is a Hardy-Littlewood system if the number of integer points on the corresponding variety restricted to a box satisfies the asymptotic behaviour predicted by the classical circle method. As a second application, we improve on a theorem of W. M. Schmidt which states that a system of homogeneous forms of same degree is a Hardy-Littlewood system as soon as the so called h-invariant of the system is sufficiently large. In this direction we generalise previous improvements of R. Dietmann on systems of quadratic and cubic forms to systems of forms of general degree.

show abstract

Forms representing forms and linear spaces on hypersurfaces

Cited by 20 publications

References 27 publications

Quadratic forms and systems of forms in many variables

Quadratic forms and systems of forms in many variables

On the number of linear spaces on hypersurfaces with a prescribed discriminant

A Variant of Weyl’s Inequality for Systems of Forms and Applications

Contact Info

Product

Resources

About