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1. CMB 2003 (vol 46 pp. 229)
Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials |
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)
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 |