Polynomial of degree N with integer coefficient for a given root. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:51:22Z http://mathoverflow.net/feeds/question/78650 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78650/polynomial-of-degree-n-with-integer-coefficient-for-a-given-root Polynomial of degree N with integer coefficient for a given root. Arif 2011-10-20T07:25:39Z 2011-10-20T18:16:49Z <p>Is it possible to construct a polynomial of degree <code>N</code>, with all of them as integer coefficient have a <code>root</code> as the given value. The root value provided is not necessarily a rational number.</p> <p>For example, if the root is <code>28.552622898861801</code> we can have a polynomial of degree 10 whose one root will be the given value.</p> <p><code>10000 x^10 - 280000 x^9 - 150000 x^8 - 220000 x^7 - 40000 x^6 - 790000 x^5 - 160000 x^4 - 320000 x^3 - 270000 x^2 - 250000 x - 251271 = 0</code> </p> <p><a href="http://www.wolframalpha.com/input/?i=10000+x%5E10+-+280000+x%5E9+-+150000+x%5E8+-+220000+x%5E7+-+40000+x%5E6+-+790000+x%5E5+-+160000+x%5E4+-+320000+x%5E3+-+270000+x%5E2+-+250000+x+-+251271+%253D+0" rel="nofollow">Wolfram alpha link for this equation</a></p> http://mathoverflow.net/questions/78650/polynomial-of-degree-n-with-integer-coefficient-for-a-given-root/78688#78688 Answer by Max Alekseyev for Polynomial of degree N with integer coefficient for a given root. Max Alekseyev 2011-10-20T18:08:02Z 2011-10-20T18:16:49Z <p>The problem can be solved by running some Integer Relation algorithm (e.g., PSLQ) on the numbers \$1, r, r^2, \dots, r^N\$ where \$r\$ is a given root.</p> <p>See <a href="http://en.wikipedia.org/wiki/Integer_relation_algorithm" rel="nofollow">http://en.wikipedia.org/wiki/Integer_relation_algorithm</a></p> <p>For example, here is computation in PARI/GP which gives a better result than the polynomial shown in question: </p> <p>? r = 28.552622898861801; algdep(r,10)</p> <p>%1 = 3*x^10 + 38*x^9 - 3695*x^8 + 4582*x^7 + 3016*x^6 + 1435*x^5 + 4552*x^4 - 1219*x^3 - 9920*x^2 - 2402*x + 3087</p> <p>? subst(%1,x,r)</p> <p>%2 = -2.7334689816478450022 E-24</p>