let' s define 'height' of a polynomial the max of the absolute value of its coefficients.
Do exist P(x) and Q(x) TWO polynomials at integer coefficient with heights equal or greater than 2011, whose product is a polynomial with height =1?
I have tried with cyclotomic polynomials, that we know can have any coefficient among the integers, and whose product of convenient cyclotomic gives x^n-1 (whoe height is 1), but I cannot demonstrate that I can find exactlky TWO polynomials whith height >=2011 whose product is a polyn. with height=1.
has anyone a clue onhow to solve with cyclotomic or with any other method?
Okay, here is the idea. Given a positive integer N, we want to find polynomials P(x), Q(x), with integer coefficients, each having height at least N, such that P(x)Q(x) has height 1.
Even for N = 2 this takes a bit of thought. Start with the polynomial
which clearly has height 1. Now by repeatedly factorising the difference of two squares, write that polynomial as
Take and Then each of P(x), Q(x) has height at least 2, and their product has height 1.
Now do the same sort of thing for N = 3, looking at the polynomial Once you have sorted that one out, you will see how to do it for any N. (It's really a very simple-minded construction, nothing as sophisticated as cyclotomic polynomials!)
brilliant! definetly very simple!!!!! thanks a lot!!!!!! the trick is to multiply with such polyn with degree such that the product gives degrees that whose terms do not sum and such that the different facorization gives the desired coefficients!!!!
the question I would like to ask is: how one can have this 'enlightment' to try this factorization? experience? chance? trials on basic polynomials?...