Counting and Testing Dominant Polynomials

Abstract

In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In particular, our results imply that the probability that what is known as the dominant root assumption holds for a random monic polynomial with integer coefficients tends to 1 in some setting. However, for arbitrary integer polynomials it does not tend to 1. For instance, the proportion of dominant quadratic integer polynomials of height H among all quadratic integer polynomials tends to (41 + 6log 2)/72 as H → ∞. Finally, we design some algorithms to test whether a given polynomial with integer coefficients is dominant without finding the polynomial’s roots.

Keywords:

• Dominant polynomial
• linear recurrence sequence
• Sturm’s theorem
• Bistritz stability criterion

2000 AMS Subject Classification::

• Primary 11C08
• Secondary 11B37, 11R06