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ALGEBRAIC NUMBERS WITH BOUNDED DEGREE AND WEIL HEIGHT

  • ARTŪRAS DUBICKAS (a1)

Abstract

For a positive integer $d$ and a nonnegative number $\unicode[STIX]{x1D709}$ , let $N(d,\unicode[STIX]{x1D709})$ be the number of $\unicode[STIX]{x1D6FC}\in \overline{\mathbb{Q}}$ of degree at most $d$ and Weil height at most $\unicode[STIX]{x1D709}$ . We prove upper and lower bounds on $N(d,\unicode[STIX]{x1D709})$ . For each fixed $\unicode[STIX]{x1D709}>0$ , these imply the asymptotic formula $\log N(d,\unicode[STIX]{x1D709})\sim \unicode[STIX]{x1D709}d^{2}$ as $d\rightarrow \infty$ , which was conjectured in a question at Mathoverflow [https://mathoverflow.net/questions/177206/].

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This research was funded by the European Social Fund according to the activity Improvement of researchers qualification by implementing world-class R&D projects of Measure no. 09.3.3-LMT-K-712-01-0037.

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ALGEBRAIC NUMBERS WITH BOUNDED DEGREE AND WEIL HEIGHT

  • ARTŪRAS DUBICKAS (a1)

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