The article presents some results concerning H-bases and theirs applications in multivariate interpolation. We derived the space of reduced polynomials with respect to a particular inner product. We made some connections with least interpolation and presented two application of the connection between spaces of reduced polynomials modulo a H-basis and spaces of ideal interpolation. 2000 Mathemat...

For a quartic primitive CM field K, we say that a rational prime p is evil if at least one of the abelian varieties with CM by K reduces modulo a prime ideal p|p to a product of supersingular elliptic curves with the product polarization. We call such primes evil primes for K. In [GL], we showed that for fixed K, such primes are bounded by a quantity related to the discriminant of the field K. ...

The existence of exact upper bounds for increasing sequences of ordinal functions modulo an ideal is discussed. The main theorem (Theorem 18 below) gives a necessary and suucient condition for the existence of an exact upper bound f for a < I-increasing sequence f = hf : < i On A where > jAj + is regular: an eub f with lim inf I cff(a) = exists if and only if for every regular 2 (jAj;) the set ...

We establish new classes of Ratliff-Rush closed ideals and some pathological behavior of the Ratliff-Rush closure. In particular, Ratliff-Rush closure does not behave well under passage modulo superficial elements, taking powers of ideals, associated primes, leading term ideals, and the minimal number of generators. In contrast, the reduction number of the Ratliff-Rush filtration behaves better...

A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown to be SOS modulo its gradient ideal, provided the gradient ideal is radical or the polynomial is...