Strong positivity results for polynomials of bounded degree.

• Main Author: Kelm, Alan Walter.
• Other Authors: Handelman, David
• Format: Others
• Published: University of Ottawa (Canada) 2009
• Subjects: Mathematics.
• Online Access: http://hdl.handle.net/10393/6614; http://dx.doi.org/10.20381/ruor-14926

Given a polynomial f possibly having negative coefficients, and a polynomial P having only positive coefficients, consider the problem of determining whether, $${\rm for\ some}\ n,\ {\rm the\ product}\ P\sp{n}f\ {\rm has\ no\ negative\ coefficients}.\eqno(*)$$ Generalizations of the positivity problem can be made in several ways. De Angelis (DA1) has dropped the condition that P be positive, requiring instead that $\vert P(re\sp{i\theta})\vert0$ and Another generalization involves permitting P to vary. That is, to replace $P\sp n$ by a product $P\sb1P\sb2P\sb3\cdot\cdot\cdot P\sb n$, of positive polynomials. This was first considered in (H3, Appendix C), which gives some results for the case of several variables. The appropriate generalization of the problem is to take a sequence $\{P\sb i\}$ of positive polynomials and to ask whether, for all eligible polynomials f, $${\rm for\ every}\ k\ {\rm in}\ {\bf N}\ {\rm there\ is\ some}\ n,\ {\rm for\ which\ the\ product}$$ P$\sb{k+1}P\sb{k+2}\cdot\cdot\cdot P\sb{k+n}\cdot f$ has no negative coefficients.(**) In (BH) the problem (**) is studied extensively for the case of polynomials in a single variable x. The sequence $\{P\sb i\}$ of polynomials is said to be strongly positive if (**) holds for all f for which $f\vert\sb{(0,\infty)}$ is strictly positive. It is frequently convenient to permit the polynomials $P\sb i$ and f to be Laurent polynomials; that is, polynomials admitting both positive and negative powers of the indeterminate x. Before studying strong positivity, it is important to be familiar with what is called the fluctuation of a Laurent polynomial. If $P=\sum\sb ip\sb ix\sp i,$ then the fluctuation ${\cal F}(P)$ of P is the number $\sum\sb i\vert p\sb i-p\sb{1+1}\vert\over \sum\sb ip\sb i$. It measures the differences between adjacent coefficients of P as a fraction of the total mass of P. A necessary condition for a sequence $\{P\sb j\}$ to be strongly positive is that $${\cal F}(P\sb{k+1}P\sb{k+2}\cdot\cdot\cdot P\sb{k+n})\to 0,\quad {\rm as}\ n\to\infty,\ {\rm for\ every} k.\eqno(+)$$ We end chapter 5 by giving a few circumstances in which we can guarantee strong positivity even though $\sum\sb i{\cal P}(P\sb i)$ is finite. These are generally cases in which lockings into products will readily yield divergent persistence. We also note that these cases tend to require that the dominant coefficients have relatively prime spacing. Failing this, the sequence is often not even zero-fluctuation sequence. An underlying topic which surfaces at various points throughout the thesis is strong unimodality of a Laurent polynomial (also known as log-concavity). In chapter 7 we prove a generalization of a theorem due to Odlyzko and Richmond (OR). They proved that under suitable conditions on the positioning of the nonzero coefficients of P, all large powers $P\sp n$ will be strongly unimodal. Our generalization proves the strong unimodality of all products $P\sb1P\sb2\cdot\cdot\cdot P\sb n$ for n sufficiently large, subject to appropriate conditions on the sequence $\{P\sb i\}.$ In chapter 6 we prove another strong unimodality result, this time for sums of products or linear polynomials. The result is phrased as strong unimodality of various cross-sections of a two-dimensional triangular grid of coefficients. Results of this type come into play when seeking to estimate coefficients of powers of certain univariate polynomials. (Abstract shortened by UMI.)