Multinomial Points

We describe the integer-valued functions which call arise as the image of an integral coefficient polynomial in k variables when it or a related power series is evaluated at k-tuples of integers from the domain {0, 1......, n-1} and also when it. is evaluated at, k-tuples of natural numbers. There is an interesting duality between the coefficients and the values of such polynomials and power series, which has applications in number theory. The techniques include expanding Lagrange interpolation polynomials to power series with respect to a basis for multivariable polynomials, called the integral root basis, and constructing higher dimensional analogs of Pascal's infinite matrix.