Abstract

Let $\mathbf{f}=(f_{1},\ldots ,f_{R})$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_{j}(x_{1},\ldots ,x_{n})=0~(1\leqslant j\leqslant R)$ satisfies a general local to global type statement, and has a solution where each coordinate is prime. In fact we obtain the asymptotic formula for number of such solutions, counted with a logarithmic weight, under these conditions. We prove the statement via the Hardy–Littlewood circle method. This is a generalization of the work of Cook and Magyar [‘Diophantine equations in the primes’, Invent. Math.198 (2014), 701–737], where they obtained the result when the polynomials of $\mathbf{f}$ all have the same degree. Hitherto, results of this type for systems of polynomial equations involving different degrees have been restricted to the diagonal case.

Highlights

  • Let d 1, and let f = be a system of polynomials in Z[x1, . . . , xn], where f = ( f,1, . . . , f,r ) is the subsystem of degree polynomials of f (1 d)

  • We present Theorem 8.1 in Section 8, where we obtain the asymptotic formula for the number of prime solutions, counted with a logarithmic weight, instead of solutions whose coordinates are all prime powers as in Theorem 1.2

  • As in the case for integer points, there are significant challenges to be overcome in generalizing the result on prime solutions of polynomial equations of equal degree to handle arbitrary systems

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Summary

Introduction

(These are slightly different from the local conditions described in the paragraph after Theorem 1.2.) As stated in [3], ‘Birch’s original result needed the forms all to have the same degree, and there is a significant technical problem in extending the method to the general case.’. It is required in Theorem 1.1 that the polynomials all have the same degree. As in the case for integer points, there are significant challenges to be overcome in generalizing the result on prime solutions of polynomial equations of equal degree to handle arbitrary systems. There should be no ambiguity since we are defining these notations as they come up

Regularization lemmas
Decomposition of forms
Hardy–Littlewood circle method: minor arcs
Technical estimates
Hardy–Littlewood circle method: major arcs
Conclusions and further remarks
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