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A Goldbach theorem for polynomials of low degree over odd finite fields. (English) Zbl 0491.12017

A monic polynomial \(M(x)\) of degree \(r\) over the finite field \(\mathbb F_q\) \((q\) odd) is called a 3-primes polynomial if it can be written as the sum of three monic irreducible polynomials, one of degree \(r\) and the others of degree less than \(r\). In this paper it is proved that every monic polynomial of degree \(2, 3, 4, 5\), or \(6\) is a 3-primes polynomial, and also that if \(\deg(M) = 7\) and if \(q\ge 207\), then \(M\) is a 3-primes polynomial.
This result supplements a theorem of D. R. Hayes (unpublished) which asserts that for every degree \(r\ge 5\), there exists a \(q_r\) (depending only on \(r\) and decreasing as \(r\) increases) such that if \(q\ge q_r\), then every monic polynomial of degree \(r\) over \(\mathbb F_q\) is a 3-primes polynomial. It can be calculated from the proof of the Hayes theorem that \(q_5\le 6, 340, 567\), \(q_6\le 5, 297\), \(q_7\le 479\), and \(q_8\le 137\); so the theorem of this paper supplies information about the existence of 3-primes polynomials not supplied by the Hayes theorem.
The cases \(r=2,3,4\) and \(5\) are handled by relatively elementary applications of orthogonal geometry over finite fields. For the cases \(r=6\) and \(7\), however, an asymptotic technique (suggested by S. D. Cohen [Acta Arith. 17, 255–271 (1970; Zbl 0209.36001)]) using Artin non-abelian \(L\)-functions, A. Weil’s Generalized Riemann Hypothesis for function fields, and the Chebotarev Density Theorem results in the calculation that in fact \(q_8\le 19\) and \(q_7\le 207\). Finally the cases \(r= 6,\ q =3\) through \(17\) are handled by the computer. It is observed that a complete solution to the “3-primes conjecture for polynomials over finite fields” is now probably an executable computer calculation way.

MSC:

11T06 Polynomials over finite fields
11P32 Goldbach-type theorems; other additive questions involving primes
51E99 Finite geometry and special incidence structures
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors

Citations:

Zbl 0209.36001
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