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1. CMB 2012 (vol 56 pp. 844)
On the Average Number of Square-Free Values of Polynomials We obtain an asymptotic formula for the number
of square-free integers in $N$ consecutive values
of polynomials on average over integral
polynomials of degree at most $k$ and of
height at most $H$, where $H \ge N^{k-1+\varepsilon}$
for some fixed $\varepsilon\gt 0$.
Individual results of this kind for polynomials of degree $k \gt 3$,
due to A. Granville (1998),
are only known under the $ABC$-conjecture.
Keywords:polynomials, square-free numbers Category:11N32 |
2. CMB 2007 (vol 50 pp. 409)
Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields We show that, for most of the elliptic curves $\E$ over a prime finite
field
$\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number
field containing the endomorphism ring of $\E$ over $\F_p$
is sufficiently large.
We also obtain an asymptotic formula for the number of distinct
quadratic number fields generated by the endomorphism rings
of all elliptic curves over $\F_p$.
Categories:11G20, 11N32, 11R11 |
3. CMB 2003 (vol 46 pp. 71)
The Number of Fields Generated by the Square Root of Values of a Given Polynomial The $abc$-conjecture is applied to various questions involving the
number of distinct fields $\mathbb{Q} \bigl( \sqrt{f(n)} \bigr)$, as we vary
over integers $n$.
Categories:11N32, 11D41 |