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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