Diophantine equations in semiprimes

Diophantine equations in semiprimes, Discrete Analysis 2019:17, 21 pp. This paper considers the problem of finding integer solutions to integral polynomial equations of the form $$f(x_1,\dots,x_n)=0\qquad\qquad (*)$$ with the condition that each coordinate $x_i$ has a 'small' number of prime factors. This is a very general problem, and even without any added restrictions on the coordinates, determining whether there exists an integral solution to $(*)$ is known to be an impossible task (this is a consequence of the work on Hilbert's 10th problem). However, it was shown by Bryan Birch that under some suitable nondegeneracy conditions on $f$ one can give local-to-global general results. Loosely, these conditions can be interpreted as the statements that $f$ is a function in sufficiently many variables (as a function of the degree) and also that there are no divisibility obstructions to $(*)$. More precisely, the polynomial $f$ needs be large with respect to a certain notion of rank and $(*)$ needs to have solutions over the $p-$adics for all primes $p$. This lower bound on the rank is what is meant by 'sufficiently many variables', and in Birch's result the bound is roughly $2^d$, where $d$ is the degree of the polynomial. A recent paper of Brian Cook and Ákos Magyar proves a variant of Birch's result that counts solutions to $(*)$ in which all the coordinates are primes. The style of that result is in line with Birch's result, but they need far more variables than the $2^d$ benchmark set by Birch. In a slightly different direction, a more recent paper of Magyar and Tatai Titichetrakun gives a Birch-type result on solutions to $(*)$ in which the coordinates are required to be _almost_ primes, which means that they have a small number of prime factors. Interestingly, their result of matches the results of Birch from the point of view of the 'sufficiently many variables' requirement. On the other hand, the notion of an almost prime adds a new quantitative parameter -- how small a number of prime factors we can take. In the paper of Magyar and Titichetrakun, the number of prime factors needed for the main result is $384n^{3/2}d(d+1)$. In particular, it depends on both the degree and the number of variables. It is believed that one can obtain prime solutions -- that is, solutions where there is just one prime factor -- so there is expected to be a great deal of room for improvement. The main result of the current paper does indeed prove a drastic reduction on the number of prime factors needed, right down to 2. A number with exactly two (not necessarily distinct) prime factors is called a _semiprime_, so the paper finds solutions to $(*)$ for which the coordinates are semiprimes. This improvement does come at cost in the required number of variables, but it turns out to be a somewhat modest one: instead of roughly $2^d$ variables, the proof requires roughly $4^d$ variables. The argument is a very clever application of the circle method of Hardy and Littlewood combined with recent work of Damaris Schindler on integral solutions of $(*)$ when $f$ is a bihomogeneous polynomial. To get at the result on solutions in semiprimes the author passes through an intermediate result (Theorem 2.1) which shows up in an analogue of Schindler's work and gives a weighted count of prime solutions to $(*)$ when $f$ is bihomogeneous. This is carried out by an application of the circle method of Hardy and Littlewood (which is somewhat similar in style to an argument of Lilu Zhao). With the result on prime solutions for bihomogeneous equations in hand, finding solutions to $f(x)=0$ in semiprimes is a tractable problem, as it is equivalent to finding solutions to the equation $f(x_1y_1,...,x_ny_n)=0$ in primes $x_1,...,y_n$. A notable feature of the methods used in this paper is what is missing: there is no application of sieve techniques. Almost primes usually arise as a substitute when one faces a Diophantine problem in the primes that is out of reach of current methods. For example, the binary Goldbach conjecture is not approachable with current methods, but Chen Jingrun famously solved this question in semiprimes. There are numerous other examples of results of this flavour, but it is very unusual for them not to rely on some form of sieve techniques.

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