With the help of volumetric formula (V=1/2x2x2) of a spheroid that U 2+V 2+W 2+Z 2≤x in four dimensional space, we have got an asymptotic formula A(x) = 1/2 x2x2 + O(x3/2) where A(x) is the number of integral points in the spheroid.
For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation.
In particular, we prove an integral formula for the degree of an ample divisor on a variety of complexity 1, and apply this formula to computing the degree of a closed 3-dimensional orbit in any SL2-module.
Essential Dimension of Algebraic Groups and Integral Representations of Weyl Groups
Let k be an integral domain, n a positive integer, X a generic n × n matrix over k (i.e., the matrix (xij over a polynomial ring k[xij] in n2 indeterminates xij), and adj(X) its classical adjoint.
We show that the structure of a block outside the critical hyperplanes of category O over a symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules.