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Search: MSC category 11Y99 ( None of the above, but in this section )

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1. CMB 2003 (vol 46 pp. 229)

Lin, Ke-Pao; Yau, Stephen S.-T.
Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials
Recently there has been tremendous interest in counting the number of integral points in $n$-dimen\-sional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in $n$-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous $n$-dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.

Categories:11B75, 11H06, 11P21, 11Y99

2. CMB 1998 (vol 41 pp. 125)

Boyd, David W.
Uniform approximation to Mahler's measure in several variables
If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure of $f$, $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus $\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy $-d2^{-n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number of arithmetic operations from the coefficients of $f$ this also demonstrates an effective (but impractical) method for computing $M(f)$ to arbitrary accuracy.

Categories:11R06, 11K16, 11Y99

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