The distance to an irreducible polynomial, II

Abstract

Abstract. P. Turán asked if there exists an absolute constant C such that, where L(·) denotes the sum of the absolute values of the coefficients. We show that C = 5 suffices for all integer polynomials of degree at most 40 by investigating analogous questions in Fp [x] for small primes p. We also prove that a positive proportion of the polynomials in F 2 [x] have distance at least 4 to an arbitrary irreducible polynomial.

Is Every Polynomial with Integer Coefficients Near an Irreducible Polynomial?

“…The sum of the degrees of the moduli used in the construction of f (x) in F p [x] is in general ≤ 19, and this is enough to show the existence of an f (x) as stated earlier of degree ≤ 18. The density argument for the result in F p [x] as before follows along the lines of[6], and one can in fact deduce that asymptotically at least 1/ p 19 of the f (x) ∈ F p [x] are a distance ≥ 3 from an irreducible polynomial.Before ending, we note that the simple looking polynomial f (x) = 5x 5 + 8x 4 + 2x 3 + 9x 2 + 10x has the property that f (x) has distance ≥ 3 from every irreducible polynomial inF 17 [x]. Thus, the existence result for polynomials of degree ≤ 18 in F p [x] that are a distance ≥ 3 from every irreducible polynomial in F p [x] is not sharp, at least for all primes p. In fact, one can show that the bound 18 can be replaced by ≤ 8 for every prime p ≡ 1 (mod 8).…”

“…x + 1 0 x] and, hence, one may take C = 5 for every polynomial of degree ≤ 24 in Turán's problem. These computations have been extended further using different approaches by Gilbert Lee, Frank Ruskey and Aaron Williams [8], Michael J. Mossinghoff [9], and Mossinghoff and the author [6]. At this point, we know that one may take C = 5 for polynomials up to degree 40.…”

“…To show that a positive density of the polynomials in F 3 [x] are a distance ≥ 3 from an irreducible polynomial, it suffices to show that a positive density of the polynomials satisfy the above congruences. This will be clear to some of our readers, and we appeal to[6] for details, where a similar argument is done over F 2[x]. The more general result for an arbitrary odd prime p can be done using a very similar argument.…”

The distance to square-free polynomials

“…Moreover, there are only 37 positive integers r ≤ 10 6 for which the quotient Φ(r)/r is at least 2. We list them as follows: 2,4,6,8,9,10,12,13,16,18,20,24,25,26,32,36,40,42,44,48,49,50,72,73,74,80,84,96,97,120,121,144,145,240,241, 242, 288.…”

“…In [1], Banerjee and Filaseta improved the above upper bound towhere c 0 is an effectively computable absolute constant. In addition, using computational strategies, it has been confirmed in [3,4,9,12,13] that if f ∈ Z[x] has degree d ≤ 40, then there exists an irreducible polynomial g ∈ Z[x] with deg g = d and L(f − g) ≤ 5. On the other hand, although the trivial example f (x) = x 3 shows that C ≥ 2, it is not known that the optimal constant C should be strictly greater than 2.In this paper, we consider a variant of Turán's problem, where "irreducible polynomial g" is replaced by "square-free polynomial g".…”