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  The previous posting is now superseded.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).

Extending Integer Polynomial Pseudo-Derivative Results

Last time, I demonstrated my own definition of an integer polynomial pseudo-derivative and how it can be used to construct the results of the polynomial using only sums.  Now I wish to look closer at some of the columns in the pseudo-derivative tree of the same polynomial: p(x) = ax^3 + bx^2 + cx + d.

Looking back at the previous post, there are four columns in the result portion of the pseudo-derivative tree (the supplied “x” variable is not counted as a result column).

Column 4 is the final nonzero pseudo-derivative for this polynomial, and the column has only 6a in every entry.  Column 3 is more interesting, as it begins with 6a+2b and each successive entry is increased by 6a.  Column 2 is the one I wish to focus on now, and in particular the “a” term of each entry.  After reducing to just the “a” term, the column looks like this:





The differences between these terms have already been demonstrated, but now let’s look at the similarities.  In particular, subtract a from all terms, then divide all terms by 6a:





The term after 37a is 61a, which reduces to 10 in the final listing.  This further reinforces what is probably obvious:  this sequence is “diagonal row 2” of Pascal’s Triangle, with generating formula k(k+1)/2, which is also the famous set of Triangle numbers.  In fact, column 3 corresponds to “diagonal row 1” of Pascal’s Triangle, and if p(x) were degree 4, we would have a column which corresponds to “diagonal row 3” of Pascal’s Triangle.

This correspondence continues through the entire range of Pascal’s Triangle given polynomials of increasing degree.