Counting degenerate polynomials of fixed degree and bounded height

Artūras Dubickas; Min Sha

doi:10.1007/s00605-014-0680-9

Counting degenerate polynomials of fixed degree and bounded height

Artūras Dubickas, Min Sha^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

1 Downloads (Pure)

Abstract

In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree n ≥2 and height bounded by H ≥2. The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.

Original language	English
Pages (from-to)	517-537
Number of pages	21
Journal	Monatshefte fur Mathematik
Volume	177
Issue number	4
DOIs	https://doi.org/10.1007/s00605-014-0680-9
Publication status	Published - 25 Aug 2015
Externally published	Yes

Keywords

Degenerate polynomial
Linear recurrence sequence
Mahler measure
Resultant

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10.1007/s00605-014-0680-9

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AB - In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree n ≥2 and height bounded by H ≥2. The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.

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KW - Linear recurrence sequence

KW - Mahler measure

KW - Resultant

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Counting degenerate polynomials of fixed degree and bounded height

Abstract

Keywords

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