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I am wondering what, if anything, is known about the characteristic polynomials of integer symmetric matrices. I believe I read somewhere that not every polynomial with integer coefficients can be a characteristic polynomial for an integer matrix (correct me if I am wrong). What I would like to be true is that basically there aren't really any constraints in some sense.

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Yoav... complex? That's quite a restriction... :P – darij grinberg Jan 25 '12 at 22:36
He means that the eigenvalues of a real symmetric matrix are real. – Tom Goodwillie Jan 25 '12 at 22:43
Think of a simple case: $2\times 2$ symmetric matrices with trace zero. The question becomes, which integers may be expressed as the sum of two squares? The answer is well-known, and yes there are constraints! – Tom Goodwillie Jan 25 '12 at 22:56
Some information is in the paper E. Bender and N. P. Herzberg, Linear and Multilinear Algebra 2 (1974), 173--178. – Richard Stanley Jan 25 '12 at 23:39
If $x^2+bx+c$ is the characteristic polynomial of a (two by two) symmetric matrix then the discriminant $b^2-4c$ is a sum of two squares. This is valid over any commutative ring. The converse holds when $2$ is invertible, and also if the ring is $\mathbb Z$. – Tom Goodwillie Jan 26 '12 at 2:54

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