# How prove this polynomial inequality from a book

Question:

Let $$f(X)=(X-x_{1})(X-x_{2})\cdots (X-x_{n})$$ be an irreducible polynomial over the field of rational numbers,with integer coefficients and real zeros.

Prove that $$\prod_{1\le i<j\le n}|x_{i}-x_{j}|\ge\dfrac{n^n}{n!}$$

This result is fell interesting, and This problem is from problem book, and this problem is name Siegel created it, and I searched sometimes and can't find it.and can't solve this problem, can someone help me?

• See Siegel's book "Lectures on the geometry of numbers" (pages 27 and 28). – Lucia Dec 12 '14 at 12:32
• Here is Google Books link to the reference given by Lucia. I would be quite curious to know the name of the book where you saw this problem. – Martin Sleziak Dec 12 '14 at 13:20
• @MartinSleziak,is from ：Problems From the Book by Titu Andreescu Page 312 – math110 Dec 16 '14 at 16:20
• Is this perhaps related to the Van der Waerden conjecture (now theorem) on the end of this page? en.wikipedia.org/wiki/Doubly_stochastic_matrix – Per Alexandersson Dec 16 '14 at 20:38
• Not sure why there is a bounty when Lucia already gave an answer; perhaps another place to look would be Goldman's The Queen of Mathematics in which you will find the same result. See Chapter 22 pp. 461-462 (here) in which the result is discussed as a special case of Minkowski's proof of a conjecture due to Kronecker. – Benjamin Dickman Dec 23 '14 at 6:19