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I have been stuck with this question for… quite sometime. Implications are mentioned after my question.

Is it true that if two real polynomials $P(x), Q(x)$, have their product equal to a 0-1 polynomial (e.g. $1+x+x^5+x^{42}$), and their coefficients are assumed to be non-negative, and are both monic, then these coefficients are also 0's and 1's?

I can prove it when the product is a reciprocal polynomial (a textbook case is $1+x+x^2+x^3+\cdots+x^n$), but I cannot manage to drop this condition.

Beware that there may exist factors with negative coefficients, the assumption is that both $P, Q$ do not.

This would explain why Linear Programming in the real fields seems to always find 0-1 factorizations in some tiling problems in dim. 1 (I know this has been recently disproved in higher dimension). This is a highly rewarding occurrence but it would be nice to understand why or at least to prove it…

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    $\begingroup$ You probably need an extra assumption, like assuming that $P$ and $Q$ are monic; otherwise, take $P(x)=2x$, $Q(x)=x/2$. $\endgroup$ – W-t-P Aug 16 '19 at 14:40
  • $\begingroup$ Of course you're right, forgot to mention it. I correct the question. $\endgroup$ – Emmanuel Amiot Aug 17 '19 at 15:43
  • $\begingroup$ Note that I've asked this on MO: mathoverflow.net/questions/339137/… $\endgroup$ – Sil Aug 25 '19 at 11:38
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Observation:

Suppose that $P(x)Q(x) = M(x)$, where $M$ is a $0-1$ polynomial and $P$, $Q$ are polynomials with non-negative coefficients as in your question. Define the following property of this product:

Property 1: For each term ($x^n$ say) in $M$, there is unique pair of terms in $P$ and $Q$ which uniquely multiply together to make this term.

Firstly, it seems that the conjecture is easy to prove with the assumption that property 1 holds.

Secondly, if the conjecture does hold (so that $P$ and $Q$ are necessarily both 0-1) then property 1 must hold for the product (i.e. every product).

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(Not an answer, but too big for a comment.)

Here is a nice probability problem.

Suppose we have two six-sided dice, with faces numbered $1,\dots,6$. But these are not fair dice, they are weighted. That is, probability of outcomes $1,\dots,6$ for the first die are some nonnegative numbers $a_1,\dots,a_6$, respectively, and for the second die $b_1,\dots,b_6$. Can we fix these weights somehow so that, when the two dice are rolled, all outcomes $2,\dots,12$ for the sum are equally likely?

Answer: no. You can probably find a solution on line.

Algebraically, it means:
The polynomial $\sum_{j=0}^{10} x^j$ cannot be factored as $(a_1+a_2x+\cdots+a_6x^5)(b_1 +b_2x+\cdots+b_6x^5)$ with all coefficients nonnegative.

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