$f_{1},f_{2},\dots,f_{n}$ are polynomials with integer coefficients. Prove there exist a reducible $g(x)$ with integer coefficients that $f_{1}+g,f_{2}+g,\dots,f_{n}+g$ are irreducible.
Problem
Source: Iran 2003
Tags: algebra, polynomial, number theory unsolved, number theory
sam-n
04.04.2004 18:26
I have posted it recently but no one didn't send solution http://www.mathlinks.ro/viewtopic.php?t=4618
Myth
04.04.2004 22:25
Lemma. Suppose h(x)=x<sup>m</sup>-Bx<sup>m-1</sup>+a<sub>2</sub>x<sup>m-2</sup>+...+a<sub>1</sub>x+a<sub>0</sub>, a<sub>0</sub><>0 and B>4(m+|a<sub>m-2</sub>|+...+|a<sub>0</sub>|). Then h(x) is irreducible over Z[x]. Proof. Let us show h(z) has exactly one root outside of |z|<1. Since h(1)<0 and h(+\infty)=+\infty ==> h(x) has one real root on (1,+\infty]. Suppose u is root of h(z) and |u| \geq 1. We have 0=|h(u)|=|u<sup>m-1</sup>(u-B)+a<sub>2</sub>u<sup>m-2</sup>+...+a<sub>1</sub>u+a<sub>0</sub>| \geq |u|<sup>m-1</sup>(|B-u|-|a<sub>2</sub>|-...-|a<sub>1</sub>|-|a<sub>0</sub>|) ==> |u-B| \leq B/4 ==> Re u \geq 3B/4. Let u<sub>1</sub>, u<sub>2</sub>,...,u<sub>m</sub> be all roots of h(z), |u<sub>1</sub>|,..,|u<sub>k</sub>| \geq 1, |u<sub>k+1</sub>|,...,|u<sub>m</sub>| < 1. We have B=|u<sub>1</sub>+...+u<sub>m</sub>| > |Re u<sub>1</sub>+...+Re u<sub>k</sub>|-n \geq k*3B/4-B/4 ==> k \leq 1. Now suppose h(x)=h<sub>1</sub>(x)h<sub>2</sub>(x), where h<sub>1</sub>(x),h<sub>2</sub>(x) \in Z[x]. Then h<sub>i</sub>(x)=(...)x+c<sub>i</sub>, c<sub>i</sub> is integer number and c<sub>i</sub><>0 ==> h<sub>i</sub>(x) has at least one root outside of |z|<1 (since product of all roots is c<sub>i</sub>) ==> h(z) has at least two roots outside of |z|<1 - contradiction. We can WLOG assume f<sub>i</sub>(0)<>0 for all i=1..n (otherwise consider polynomials f<sub>i</sub>(x+s) instead of f<sub>i</sub>(x) where s some integer such that f<sub>i</sub>(s)<>0 for all i=1..n). Let m=max deg f<sub>i</sub>. Then g(x)=x<sup>m+2</sup>-Bx<sup>m+1</sup> where B is sufficiently large integer number (accordingly to lemma).