Geodesic
PRIME SOLUTIONS TO POLYNOMIAL EQUATIONS IN MANY VARIABLES AND DIFFERING DEGREES
Forum of Mathematics, Sigma, Tome 6 (2018)

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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. 
@article{10_1017_fms_2018_21,
     author = {SHUNTARO YAMAGISHI},
     title = {PRIME {SOLUTIONS} {TO} {POLYNOMIAL} {EQUATIONS} {IN} {MANY} {VARIABLES} {AND} {DIFFERING} {DEGREES}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {6},
     year = {2018},
     doi = {10.1017/fms.2018.21},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2018.21/}
}
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SHUNTARO YAMAGISHI. PRIME SOLUTIONS TO POLYNOMIAL EQUATIONS IN MANY VARIABLES AND DIFFERING DEGREES. Forum of Mathematics, Sigma, Tome 6 (2018). doi: 10.1017/fms.2018.21

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