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  The previous posting is now superseded.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).


What is it that we hear every day?  Do we hear the radio, the voices from a favorite TV serial, the sounds of a commute along extended stretches of road, the familiar voices of loved ones?

What happens when that changes?  Do we experience subtle shifts in perspective?  Does the world turn upside down in protest?  If the radio is off during a particularly long commute, would that change everything?

Having recently been introduced to the first couple chapters of Born to Run, a journalist’s novel covering his experiences with a native tribe of Tamahumara in Mexico, and later finding that the tribe is turning from its roots in running from our modern society and every other influence that may cause annihilation amongst its people, I find that I wish they had never been discovered.

There is no possible way that such a culture could have truly remained both alive and hidden from the rest of the world permanently.  As the aforementioned book chronicled, the Mexican government was already seeking ways to put roads through the best portions of the Tamahumara hiding places.  Drug cartels already laid claim to much of the territory.  Eventually, their fate was almost guaranteed to be the same as that of the bushmen, and many other aboriginal cultures.

What does our modern culture offer to them?  Sugar, iPods, jeans, a sense of ownership of things of this world…  Noise.

To a Tamahumaran, I would imagine that a typical day, more than five years ago, would have consisted of a very different set of noises.  Mountain air, desert creature sounds, the feel of simple homespun cloth, the taste of iskiate and corn in every meal.  The culture of intimately knowing your neighbors, despite the long trek required for a visit.

Now, hearing news that this culture has been turning modern, wearing of jeans, eating sugar as a dietary staple, the younger generations adopting the common language of Mexico, being more receptive to outside influence, I wonder if the modernization arrived too quickly.  I wonder if anyone else regrets the losses that are so intangible, yet so central to what was.

I turned off my car radio six months ago.  I have experienced six months’ worth of relatively silent hour-long or longer commutes to job sites.  I have never experienced a greater amount of progress, innovation, or clarity of mind than the past six months.  I have (re-)invented designs for the internal combustion engine, started my own mathematics/coding blog, progressed farther than I ever thought possible along a train of mathematical thought, and become a much better husband and father.  I have begun to set aside those distractions that weighed me down, including the most significant, which was gaming.

I am not saying that total silence, total Gnosticism, total avoidance of the things that make this world our home is the best way.  I claim that research and guru-ism and the previous way of life of the Tamahumara all have things to teach us about how we should live.

In our human history, certainly for at least ~6000 years of it, there has been a great deal of silence.  Silence in the form of lack of advertising, silence in the form of lack of alternative spices, silence in the form of lack of alternative materials for clothing, silence in the form of lack of common entertainment.

150 years ago, if you wanted to see a show, listen to a musician or band, or taste a foreign cuisine, you would have to go Somewhere Else.  When you got There, the object of your journey would be the entire fulfillment of your adventure.  There would be silence along the journey, and silence on the return.  You would return to your common portion of existence.  You would have your memories of the experience, and any discussion with friends or family as reminders.

Today, you wouldn’t have to go There.  You could Download a copy of the song, or watch the show on YouTube.  You could get a recipe Online, and find the ingredients at a local grocery store.  You wouldn’t have to feel a bumpy road, whether through shoes, moccasins, stage-coach, or horseback.

Our modern society is not bad.  But the society we left behind has good in it as well.  We often forget the silence we have been bred into under the vast majority of our human existence.  This modern age of noises is new and foreign to our DNA.  We must not forget that as we transition ourselves and our fellow cultures into it.


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.

The Pseudo-Anti-Derivative

Like Calculus, these “finite difference” methods also apply in the reverse direction, i.e., there is also a “pseudo-anti-derivative” which is the reverse of the “pseudo-derivative” process.

Given an integer polynomial p(x), let the pseudo-anti-derivative of p be marked as ‘p(x).  The value of ‘p(x) is unknown, since we must know the value of ‘p(0) to determine it.  This fits nicely with how our other methods operate, so use ‘p(0) to denote this unknown value.  Also, let (x/k) be my representation of the binomial term “x choose k”:

‘p(x) = ‘p(0) + p(0)*(x/1) + p`(0)*(x/2) + …

To show that this is a correct “pseudo-anti-derivative,” we calculate ‘p(x+1) – ‘p(x):

‘p(x+1)-‘p(x) = p(0)(((x+1)/1) – (x/1)) + p`(0)*(((x+1)/2) – (x/2)) + …

= p(0)*1 + p`(0)*(x/1) + p“(0)*(x/2) + …

Note that ((x+1)/k) – (x/k) = (x/(k-1)).  Since this is what the above proof is based on, here is why it’s true:

((x+1)/k) – (x/k) = (x+1)x(x-1)(x-2)…(x-k+2)/k! – x(x-1)(x-2)(x-3)…(x-k+1)/k! = (x+1 – (x-k+1)) * x(x-1)(x-2)…(x-k+2)/k! = kx(x-1)(x-2)…(x-k+2)/k! = (x/(k-1)).

This form of “anti-derivative” creates polynomials with all-integer results, but does not have the restriction that all coefficients are integers.  For example, if p(x) = x + 1, p(0) = p`(0) = 1, then:

‘p(x) = ‘p(0) + x + x(x-1)/2 = ‘p(0) + x/2 + x^2/2.

Refining (Pseudo-)Derivatives of Integer Polynomials

In the past couple posts, polynomials have taken on some new presentations.  This time, I’ll construct some values of a polynomial using only its values at x=0 of p(x) and all of its pseudo-derivatives.

In particular, I’ve established that p`(x)=p(x+1)-p(x).  This means that p(x+1)=p(x)+p`(x).  This has an expansion matching Pascal’s Triangle, because p(x+2) = p(x)+2p`(x)+p“(x), and p(x+3) = p(x)+3p`(x)+3p“(x)+p“`(x).

This pattern continues across all possible values of p(x).  In fact, it is possible to express every value of p(x) as p(0) + ap`(0) + bp“(0) + … where a, b, … coefficients are the entries of the xth row of Pascal’s Triangle.

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.