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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}.
Given a negative $D>-(log X)^{log 2-delta}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of discriminant
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 prin
In his paper from 1996 on quadratic forms Heath-Brown developed a version of circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight. The weight function is assumed to
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 $(2r)^r$.
Since the study by Jacobi and Hecke, Hecke-type series have received a lot of attention. Unlike such series associated with indefinite quadratic forms, identities on Hecke-type series associated with definite quadratic forms are quite rare in the lit