1

Given a polynomial $P(x)$ of degree $m>1$: $$P(x)=a_m x^m +...+ a_k x^k +...+ a_2 x^2 + a_1 x - \alpha$$ Where $\lvert a_m \rvert >...> \lvert a_k \rvert > ... > \lvert a_1 \rvert = 1$ and $\alpha$ are integer coefficients, such that $a_k <0$ $\forall k \neq m$. What methods are there to rule out a specific rational root $ \frac {1}{\beta} >0 $ ? (It is supposed that this case that $\beta$ divides $a_m$, so the rational root theorem applies, and thus this root $ \frac {1}{\beta} $ cannot be ruled out by contradiction with the theorem). According to Sturm's Theorem, one can assert that there is a root in the interval $[0,F]$, where F is a positive number. It is known that $P(x)$ has no irrational roots.

Federico Omar
  • 11
  • 5
  • 2
    Plugging it in? – Severin Schraven Jan 15 '19 at 23:05
  • Won't work... I am looking for insights on how to tackle this problem with the given conditions, and evaluating the polynomial on the given value is impossible, because the degree of the polynomial isn't defined. i.e., the polynomial has arbitrary degree m. – Federico Omar Jan 15 '19 at 23:08
  • Well, we have to know what "is defined and what not" in the question. For my taste, i would take the reciprocal and search for possibilities to avoid some integer root. It is unclear from the post if $\beta$ is positive or general. What kind of answer is expected? What is fixed and what is moving? The $a_m$ is fixed, then $\beta$ but $m$ and all other coefficients not? – dan_fulea Jan 15 '19 at 23:46
  • @dan_fulea $\beta$ is a positive value, and $\frac{1}{\beta}$ is a candidate for rational root. All roots of P(x) are rational. deg(P(x)) is non trivial ( can be greater than 4). All the coefficients of P(x) are negative, except $a_m$. – Federico Omar Jan 15 '19 at 23:53
  • That $\beta$ is $>0$ is a new information. The fact that $P$ completely splits over $\Bbb Q$ is also new. Please rewrite the question starting with the given data. Best, introduce the objects one by one, and take care for the usual $\forall$ and $\exists$ quantifiers. Please do not introduce a polynomial, and after that the degree and the coefficients, if some $\forall$ and/or $\exists$ apply only to parts of them. Please insert all details in the posted answer, not in the comments. – dan_fulea Jan 16 '19 at 00:01
  • So let us take an example. Consider the polynomial $$P(x) = -2577992256 , x^{3} - 461728028 , x^{2} - 13348521 , x + 77$$to have a clear example in mind. Now we are searching methods that apply to this given polynomial to be able to state that $1/8$ is not a root of it? (But not working with $1/7$.) (Why not compute the value in this point?) Or we want a general fixed method, that may apply to some of the polynomials in the given family, aplicability being possible maybe when $\max_j|a_{j+1}/a_j|$, $\alpha$, $\beta$ are in some "special position"?! (Or something like this.) – dan_fulea Jan 16 '19 at 12:04

0 Answers0