The probability of relatively primepolynomials in Zpk[x]

Abstract

Let PR(m, n) denote the probability that two randomly chosen monic polynomials f , g ∈ R[x] of degrees m and n, respectively, are relatively prime. Let q = p, be a prime power. We establish an explicit formula for PR(m, 2), when R = Zq, the ring of integers mod q. Given two polynomials f(x), g(x) chosen at random, what is the probability that they are relatively prime? For a ring R, we say that two polynomials f, g ∈ R[x] are relatively prime if there is no monic polynomial of positive degree that divides both f and g. Let PR(m,n) denote the probability that two randomly chosen monic polynomials f , g ∈ R[x] of degrees m and n, respectively, are relatively prime. If R has an infinite number of elements, then PR(m,n) = 1, so we restrict our attention to finite rings R. Let R = Fq, the finite field with q elements. The formula, PFq(m,m) = 1 − 1/q was proved in [3]. When q = p = 2, Reifegerste [6] gave a combinatorial proof that PF2(m,m) = 1/2. Benjamin and Bennett [1] subsequently found a beautifully simple proof generalizing these results: Theorem 1. [1] If m,n ≥ 1, then PFq(m,n) = 1− 1 q . Theorem 1 can be generalized in at least two ways. Recently, Hou and Mullen [2] have generalized Theorem 1 by considering the problem of relatively prime polynomials in several variables over a finite field. In earlier work, Gao and Panario [4] considered the probability distribution of the greatest common divisor of l randomly chosen monic single-variable polynomials in Fq[x] with degrees n1, . . . , nl as the ni → ∞. In this paper, we restrict ourselves to single-variable polynomials and explore a different perspective. As the formula in Theorem 1 only depends on the number of elements in the field Fq, one can ask whether the same formula holds when R is another ring with q elements. For example, if R = Zq, the integers mod q, does the same formula hold? It does not, but the formula for PFq(m,n) can be viewed as a first approximation to the formula for PZq(m,n). In this paper, we prove an explicit formula for PZ pk (m, 2) for p odd. Date: June 18, 2010. 2000 Mathematics Subject Classification. 11C20, 13F20, 13B25.

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Cite this paper

@inproceedings{HAGEDORN2010ThePO, title={The probability of relatively primepolynomials in Zpk[x]}, author={THOMAS R. HAGEDORN and JEFFREY HATLEY}, year={2010} }