Do you want to publish a course? Click here

Systems of forms in many variables

91   0   0.0 ( 0 )
 Publication date 2017
  fields
and research's language is English




Ask ChatGPT about the research

We consider systems $vec{F}(vec{x})$ of $R$ homogeneous forms of the same degree $d$ in $n$ variables with integral coefficients. If $ngeq d2^dR+R$ and the coefficients of $vec{F}$ lie in an explicit Zariski open set, we give a nonsingular Hasse principle for the equation $vec{F}(vec{x})=vec{0}$, together with an asymptotic formula for the number of solutions to in integers of bounded height. This improves on the number of variables needed in previous results for general systems $vec{F}$ as soon as the number of equations $R$ is at least 2 and the degree $d$ is at least 4.



rate research

Read More

97 - Kummari Mallesham 2019
In this article, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given by Henryk Iwaniec and Ritabrata Munshi in cite{H-R}.
150 - D.R. Heath-Brown 2009
It is shown that a system of $r$ quadratic forms over a ${mathfrak p}$-adic field has a non-trivial common zero as soon as the number of variables exceeds $4r$, providing that the residue class field has cardinality at least $(2r)^r$.
177 - Robert C. Rhoades 2011
Families of quasimodular forms arise naturally in many situations such as curve counting on Abelian surfaces and counting ramified covers of orbifolds. In many cases the family of quasimodular forms naturally arises as the coefficients of a Taylor expansion of a Jacobi form. In this note we give examples of such expansions that arise in the study of partition statistics. The crank partition statistic has gathered much interest recently. For instance, Atkin and Garvan showed that the generating functions for the moments of the crank statistic are quasimodular forms. The two variable generating function for the crank partition statistic is a Jacobi form. Exploiting the structure inherent in the Jacobi theta function we construct explicit expressions for the functions of Atkin and Garvan. Furthermore, this perspective opens the door for further investigation including a study of the moments in arithmetic progressions. We conduct a thorough study of the crank statistic restricted to a residue class modulo 2.
329 - D.R. Heath-Brown 2009
A variant of Brauers induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.
In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic methods is clarified and, motivated by higher order forms, new convolution products of L-functions are introduced.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا