Just as the dictionary is only as good as itself, this post will only survive as long as those who read it give its words definition within their own minds.

Comedians have often exposed the inconsistencies within our society; things like having an invisible fence, why not have an invisible dog?

Here is another inconsistency: definition.  Is it high definition, standard definition, normal definition, medium definition, or low definition?

Advertising is a major force attempting to compel common language.  Who would ever choose low definition, or even standard definition, given the option of high definition?  Why would anyone choose a payment up front, given the choice of delayed repayments, however frequent or inconvenient?  Why would anyone choose freedom, given the option of security or luxury or peace?

It’s not surprising that “high definition” is (becoming) a way of life.  We want the best.  As humans, we strive for the best.  Advertisements seek to entice us with “the best.”  And the culture of being advertised to also influences every decision.  It isn’t just about who our friends or parents or schoolfellows or coworkers are anymore.  It’s about who advertises to us.  This has been true as long as “Mad Men” has been a reality, and probably before that.  It’s about what Hollywood advertises, as much as what our families think of that perspective.  It’s about what our politicians and neighbors with political interest advertise, as much as about the real issues at hand.

But what *is* the best?  How shall we know that we really have what is best for us?  Is it possible to quantify the idea that “now, I really have what is best for me”?  If so, why would we seek anything else?  Expense (monetary, familial, friendship, etc.)?  Laziness?

In my opinion, it is always the latter.  We, as humans, are always lazy.  It does not matter whether we are type A, type B, or type Z.  Everyone is lazy.  It is always easier to commit to something familiar than what is unfamiliar.  It is always easier to listen to the advertisement voices that have been shouting for generations than to listen to the still, small, inner voices that have been there for eternity.  It is always easier to grasp the fad and the common thread than to search, and seek, and find the truth of every contrary voice that arrives.  It is always easier to “give in” than to strive.

So what is our definition?  What does it mean to be human?  Is it reasonable to give in, and allow something that is not “the best” to be a part of our experience?  Is it possible that something better, but much more difficult, is actually worth our time and consideration?  Do we have time and consideration to offer to something or someone that might be “the best” for us?  Will we someday “arrive,” or have we been attempting to “arrive” for centuries, and as humans we can never “arrive” on our own?

If it isn’t the best, why strive for it?  If it is better than anything else you can see or imagine, why not put forth every effort to make it a reality?

Integer Polynomial Lattice

We now know that any integer polynomial specified by a vector _a_ in Z^n as

(1) p(x) = a_(n-1) x^(n-1) + a_(n-2) x^(n-2) + … + a_2 x^2 + a_1 x + a_0

can also be specified by a vector _b_ in Z^n as

(2) p(x) = b_0 + b_1 x + b_2 (x | 2) + b_3 (x | 3) + … + b_(n-2) (x | n-2) + b_(n-1) (x | n-1)

where b_i = p^[i](0), the value at zero of the i-th pseudo-derivative of p(x).  It is also possible to demonstrate integer polynomials for any given _b_ for which there is no corresponding _a_, for example, _b_ = [0, 0, 1] in Z^3, i.e., p(x) = (x | 2) = x(x-1)/2.

Now we shall construct a polynomial lattice to demonstrate some of the continuity between polynomials of degree n and polynomials of the next higher or lower degree.

Begin with a given vector _b_ in Z^n, specifying a polynomial p(x) as in (2) above.  We further specify some new notation as follows:

_b`_ (read as “vector b downshift”) is a vector shift notation with values as follows: b`_0 = b_1, b`_1 = b_2, …, b`_n-3 = b_n-2, b`_n-2 = b_n-1, b`_n-1 = 0.

_b^(-k)_ (read as “vector b chopped by k”) contains values as follows: b^(-k)_0 = b_0, b^(-k)_1 = b_1, …, b^(-k)_n-k-1 = b_n-k-1, b^(-k)_n-k = 0, b^(-k)_n-k+1 = 0, …, b^(-k)_n-2 = 0, b^(-k)_n-1 = 0

With the _b`_ notation, we will also apply the _b^[i]_ notation to denote the i-th vector-shift of _b_.

The _b`_ or _b^[i]_ and _b^(-k)_ notations may be used simultaneously as _(b^[i])^(-k)_.  The order does not matter, the last k elements will be chopped to 0 prior to the downshift.

Let p(x; _a_, b, c) be an integer polynomial in x parametrized by _a_, b and c which is specified as follows:

p(x; _a_, b, c) = a_0 + a_1 (x | 1) + a_2 (x | 2) + … + a_n-2 (x | n-2) + a_n-1 (x | n-1) + b(c | n)

We use p(x; _b_, 0, 0) as the starting point of the lattice.

First, we have r(x; y) = p(x, _b^(-1)_, b_n-1, y) is the set of polynomials such that for any given x_0, p(x_0; _b_, 0, 0) = r(x_0; x_0).  This creates one axis of our lattice.

Next, we choose a value k to parametrize a second axis as follows:

q(x; k) = p(x; _b_ + (k | 1) _(b^[1])^(-1)_ + (k | 2) _(b^[2])^(-1)_ + … + (k | k-2) _(b^[k-2])^(-1)_ + (k | k-1) _(b^[k-1])^(-1)_ + _(b^[k])^(-1)_, 0, 0)

This specification identifies q(x; k) as having the same value as r(x+k; x).

Given these definitions, it is possible to identify every value of p(x; _b_, 0, 0) using polynomials of degree n-2, namely r(x; y), and it is further possible to specify the values of each of those polynomials by the values of q(x; k).  r(x; y) is degree n-2 and q(x; k) is degree n-1, demonstrating that the two sets of polynomials cover the result set differently, but in the end they both cover exactly the same result space.