most recent 30 from http://mathoverflow.net 2013-05-26T08:42:03Z http://mathoverflow.net/feeds/question/97329 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97329/complexity-of-finding-a-rational-root-of-a-polynomial Mark Sapir 2012-05-18T17:41:25Z 2012-05-19T00:27:43Z

<p>This is inspired by <a href="http://mathoverflow.net/questions/97307/polynomials-all-of-whose-roots-are-rational" rel="nofollow">this </a> question. Let $f(x)=a_nx^n+...+a_0$ be a polynomial with rational coefficients. The sandard procedure of finding a rational root $p/q$ involves checking all $p$ that divide $a_0$ and all $q$ that divide $a_0$. This is not very complicated but involves factoring $a_0$ and $a_n$. The factoring problem is not known to be in P. If $n\le 4$, then the fact that the group $S_4$ is solvable and the well known formulas for roots of polynomials of degree $\le 4$ give easy polynomial time algorithm of finding rational roots. </p> <p><b> Question</b> Is the problem of finding a rational root of $f(x)$ in P for every $n$ (say, for $n=5$)?</p> <p><b> Update 1</b> After I posted the question, I noticed an answer by Robert Israel to the previous <a href="http://mathoverflow.net/questions/97307/polynomials-all-of-whose-roots-are-rational" rel="nofollow"> question</a> (of Joseph O'Rourke). That could give an answer to my question but I am still not sure how one can avoid factoring numbers $a_0,a_n$. </p> <p><b> Update 2 </b> Robert Israel's explanations (see his comment <a href="http://mathoverflow.net/questions/97307/polynomials-all-of-whose-roots-are-rational" rel="nofollow">here </a>) convince me that his algorithm of checking whether a polynomial has a rational root (all roots rational) runs in polynomial time. </p> <p>I removed Question 2 so that I can accept Michael Stoll's answer. I will post Question 2 as a separate question. </p>

http://mathoverflow.net/questions/97329/complexity-of-finding-a-rational-root-of-a-polynomial/97338#97338 Michael Stoll 2012-05-18T18:26:02Z 2012-05-18T18:26:02Z

<p>Didn't Lenstra, Lenstra and Lovász in their famous LLL paper prove that factorization of polynomials over $\mathbb Q$ can be done in polynomial time? You get a rational root if and only if there is a factor of degree 1, and the polynomial has only rational roots if and only if all factors have degree 1.</p> <p>Lenstra, A.K.; Lenstra, H.W.jun.; Lovász, László: <em>Factoring polynomials with rational coefficients.</em> (English) Math. Ann. <strong>261</strong>, 515-534 (1982).</p>