# Correction to Polynomial Lattice

In a previous posting, I created a “polynomial lattice” out of structures found within the Pascal Triangle.  I have found some errors and boundary gaps within the definition I created.

Rather than post differences between the two, I claim that this post has Precedence.  The previous posting is now superseded.

Given a polynomial p(x) specified by a vector _b_ in Z^n as p(x) = b_0 + b_1(x | 1) + b_2(x | 2) + … + b_n-2(x | n-2) + b_n-1(x | n-1), the integer polynomial lattice surrounding p(x) is simply specified as q(x; y) = p`(x) + p(y); r(x; z) = p(x) + p`(z).  This lattice is simple, covers all the necessary bases, and achieves exactly what is required of it:  it is a lattice interleaving one set of polynomials with another; the balance of a polynomial and its parallels and the lattice-pairing of each of these is perfect; no gaps are admitted within the given parameters of the lattice, including boundaries such as when _b_ is an almost-zero vector.

The previous lattice definition allowed for many possibilities of error, and was also incorrectly specified.  Under this definition, it is possible to have _b_ = {0, 0, …, 0, 0, 1}, and to also have a viable lattice useful for determining many properties of p(x).