Challenge to the community, Squaring of polynomials conjecture

[checkitagain]

Given polynomials of degree n > 2, such that they have the form of


$p(x) \ = \ x^n \ + \ a_1x^{n - 1} \ + \ a_2x^{n - 2} \ + \ a_3x^{n - 3} \ + \ ... \ + \ a_{n - 2}x^2 \ + \ a_{n - 1}x \ + \ a_n.$


$And \ \ all \ \ of \ \ the \ \ a_i \ \ are \ nonzero \ integers \ (which \ you \ get \ to \ choose \ for \ each \ n).$


$In \ \ terms \ \ of \ \ n, \ \ what \ \ is \ \ the \ \ greatest \ \ number \ \ of \ \ terms \ \ with$


$\ \ coefficients \ \ that \ \ are \ \ zero \ \ that \ \ [p(x)]^2 \ \ can \ \ have \ ?$


$Is \ \ it \ \ (n + 1) \ \ terms \ ?$




$\text{Examples:}$


$(x^2 + 2x - 2)^2 \ = \ x^4 + 4x^3 - 8x + 4$


$(x^3 + 2x^2 - 2x + 4)^2 = x^6 + 4x^5 + 20x^2 - 16x + 16$

 

[Simon Bridge]

By that description, the greatest number of terms that (p(x))^2 can have with zero coefficients is 2n-1

p(x)=x^n (a polynomial of order n with n-1 zero coefficients)

(p(x))^2 = x^2n is a polynomial order 2n with 2n-1 coefficients.

Of course, I have not counted a₀ in that.

Unless you mean something different by "zero coefficient".

 

[checkitagain]

By that description, the greatest number of terms that (p(x))^2 can have with zero coefficients is 2n-1

p(x)=x^n (a polynomial of order n with n-1 zero coefficients)

(p(x))^2 = x^2n is a polynomial order 2n with 2n-1 coefficients.

Of course, I have not counted a₀ in that.

Unless you mean something different by "zero coefficient".

No, if you read it again, $none$ of the terms of p(x) are to have
coefficients that are zero. I want to know a formula, if it exists,
in terms of n, such that $[p(x)]^2$ itself has the most terms
possible with coefficients that are zero.



$\text{ * * * Note: This problem is still open.}$

 

[Simon Bridge]

Ah - none of the coefficients of p(x) can be zero.
So we are trying to eliminate coefficients by making some of the initial coefficients negative.

Enumerating coefficients - s we want to maximize N₀.

So I figure the real challenge is to find a simpler proof by limiting to the case n=2.
Otherwise there is some crunching through matrices to do.

 

[Office_Shredder]

Is there a particular reason you inquire if it's n+1 terms given that both your examples only end up with n-1 non-zero terms?

As a first glance, let's just start picking the high coefficients at random and then see if we can select the low coefficients intelligently. If we fix every coefficient of x^n/2 and higher terms, then every coefficient in the square that involves x^3n/2 or a higher order term can be expressed as a linear combination of the unfixed coefficients (because to get a x^3n/2 you must involve at least one term of degree n/2 or greater). And we have n/2 unknowns, so on average you should be able to get a solution.

 

[checkitagain]

Is there a particular reason you inquire if it's n+1 terms given
that both your examples only end up with $> >$n-1 non-zero terms$< <$?

In my examples:

Wait, I noticed instead that for n = 2, the number of nonzero terms is 4, or (n + 2).

And, for n = 3, the number of nonzero terms is 5, or again (n + 2).


Another example:


n = 4

$(x^4 + 2x^3 - 2x^2 + 4x + 4)^2 = x^8 + 4x^7 + 28x^4 + 32x + 16$


But now, the number of nonzero terms is (n + 1) = 5 (instead of a predicted 6).

 

[guillefix]

$In \ \ terms \ \ of \ \ n, \ \ what \ \ is \ \ the \ \ greatest \ \ number \ \ of \ \ terms \ \ with$


[itex]\ \ coefficients \ \ that \ \ are \ \ zero \ \ that \ \ [p(x)]^2 \ \ can \ \ have \ ?

In my examples:

Wait, I noticed instead that for n = 2, the number of nonzero terms is 4, or (n + 2).

Wait I got confused, so first you are asking about the number of coefficients that are zero and then about the number of coeffeficients that aren't, which one are you looking for an expression?

 

[micromass]

Checkitagain, could you PM your solution to a mentor so this challenge can stay open??

 

[checkitagain]

I am asking this to be closed.

I PMed a mentor for that.

 

[micromass]

Closed by request of the OP.