[Paper Review] Prime solutions to polynomial equations in many variables and differing degrees

Shuntaro Yamagishi|arXiv (Cornell University)|Mar 9, 2017

Analytic Number Theory Research16 references3 citations

TL;DR

This paper establishes sufficient conditions for the existence of prime solutions to systems of polynomial equations with varying degrees using the Hardy-Littlewood circle method. It proves an asymptotic formula for the number of prime solutions, generalizing prior results that required all polynomials to have the same degree by introducing a codimension condition on the singular locus of each degree component.

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 $f_j (x_1, \ldots, x_n) = 0 \ (1 \leq j \leq R)$ 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 B. Cook and Á. Magyar, 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.

Motivation & Objective

• To extend the theory of prime solutions to polynomial equations beyond the case of uniform degree polynomials.
• To establish sufficient conditions under which systems of polynomial equations with mixed degrees admit solutions in prime numbers.
• To generalize the work of Cook and Magyar, who previously treated only systems of polynomials of the same degree.
• To quantify the number of prime solutions via an asymptotic formula with logarithmic weights.
• To introduce a codimension condition on the singular locus of each degree component as a key hypothesis.

Proposed method

• Applies the Hardy-Littlewood circle method to analyze exponential sums associated with the system of polynomials.
• Defines the singular locus $ V_{oldsymbol{F}_ u}^* $ as the set of complex points where the Jacobian matrix of the degree-$ u $ homogeneous parts has rank less than the number of equations.
• Introduces the Birch rank $ \mathcal{B}_\ell(\mathbf{F}_\ell) $ as the codimension of this singular locus to measure the non-degeneracy of the system.
• Imposes a lower bound on $ \mathcal{B}_\ell(\mathbf{F}_\ell) $ in terms of $ d $, $ r_d, \ldots, r_1 $ to ensure sufficient complexity for prime solutions.
• Uses local conditions and the circle method to derive an asymptotic formula for the number of prime solutions, weighted by logarithms.
• Employs a transference principle and estimates on exponential sums to control minor arcs and establish the main term.

Experimental results

Research questions

• RQ1 Under what conditions does a system of polynomial equations with mixed degrees admit a solution in prime numbers?
• RQ2 How can the Hardy-Littlewood circle method be adapted to handle systems with varying-degree polynomials?
• RQ3 What role does the codimension of the singular locus play in ensuring the existence of prime solutions?
• RQ4 Can an asymptotic formula for the number of prime solutions be established when the polynomials are not all of the same degree?
• RQ5 How does the result generalize the earlier work of Cook and Magyar on systems of uniform degree?

Key findings

• The system $ f_{\ell,r}(x_1,\ldots,x_n) = 0 $ for $ 1 \leq \ell \leq d $, $ 1 \leq r \leq r_\ell $ has a solution in prime numbers if the Birch rank $ \mathcal{B}_\ell(\mathbf{F}_\ell) $ is sufficiently large relative to $ d $, $ r_d, \ldots, r_1 $.
• An asymptotic formula is established for the number of prime solutions, counted with logarithmic weights, under the stated conditions.
• The result generalizes Cook and Magyar's theorem by allowing polynomials of different degrees, previously restricted to the diagonal case.
• The proof relies on controlling exponential sums via a new estimate on the singular locus and a refined application of the circle method.
• The method applies even when the system is not diagonal, extending results beyond the scope of earlier works.
• The conditions include local solubility and a quantitative bound on the codimension of the singular locus, ensuring non-degeneracy of the system.