  For positive integers d, r, and M, we consider the class of rational
functions on real d-dimensional space whose denominators are products of at
most r functions of the form 1+Q(x) where each Q is a quadratic form with
eigenvalues bounded above by M and below by 1/M. Each numerator is a monic
monomial of the same degree as the corresponding denominator. Then we form the
Banach space of countable linear combinations of such rational functions with
absolutely summable coefficients, normed by the infimum of sums of absolute
values of the coefficients. We show that for rational functions whose
denominators are rth powers of a specific 1+Q, or differences of two such
rational functions with the same numerator, the norm is achieved by and only by
the obvious combination of one or two functions respectively. We also find
bounds for coefficients in partial-fraction decompositions of some specific
rational functions, which in some cases are quite sharp.
