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CONJECTURE

Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$th column of $A$ is obtained by applying the permutation $(1,2,..,n)^{i-1}$.

Assume $A \in GL_n(Z)$, i.e $A$ with integer entries and determinant $\pm 1$ and moreover $c_0+c_1+\ldots+c_n=\pm 1$.

Then there exists one $j$ such that $c_j=\pm 1$ and $c_i=0$ for all $i$ different from $j$.

Is this conjecture true?

What if we add the assumption that $n=p$ a prime?

Thanks for any idea! Fabienne

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I think this is false.

Take the first row $A=(1,-1,1,-1,1,0,0)$.

The circulant matrix is: $$ \left(\begin{array}{rrrrrrr} 1 & -1 & 1 & -1 & 1 & 0 & 0 \\ 0 & 1 & -1 & 1 & -1 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1 & -1 & 1 \\ 1 & 0 & 0 & 1 & -1 & 1 & -1 \\ -1 & 1 & 0 & 0 & 1 & -1 & 1 \\ 1 & -1 & 1 & 0 & 0 & 1 & -1 \\ -1 & 1 & -1 & 1 & 0 & 0 & 1 \end{array}\right) $$

The determinant is $1$.

Your definition is with columns, so you may need to transpose.

Experimentally first row starting $(1,-1,1)$ followed by $n$ zeros with determinant $\pm 1$ is A047235 Numbers that are congruent to {2, 4} mod 6

Added Solution with bigger $c_i$ is first row $(-2,6,-7,6,-2)$

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    $\begingroup$ A smaller example is the circulant with first row $1,-1,1,0,0$. $\endgroup$
    – Gerry Myerson
    Commented May 27, 2014 at 11:30
  • $\begingroup$ @GerryMyerson Indeed. Working symbolically don't think counterexamples exist for n=3 or n=4. $\endgroup$
    – joro
    Commented May 27, 2014 at 11:47
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    $\begingroup$ There's a paper "Unimodular Integer Circulants" by John Cremona that studies this question in some depth: homepages.warwick.ac.uk/~masgaj/papers/unicirculant.pdf $\endgroup$
    – Jeremy Rickard
    Commented May 27, 2014 at 12:49
  • $\begingroup$ Regarding the "Experimentally" edit, for even $n$ the determinant is the product of $(1+w^3)/(1+w)$ taken over all solutions $w$ of $w^{n+3}=1$. If $\gcd(n,3)=1$, then the numerator and denominator run over the same numbers, so this product is 1. $\endgroup$
    – Gerry Myerson
    Commented May 27, 2014 at 23:39
  • $\begingroup$ Thanks a lot for all your answers. I was convinced it was true. Is there a name for a matrix like a circulant but where the i-th column is a permutation f_i of the first column and f_i not necessarily (1,2, ..,n)^i? $\endgroup$
    – Fabienne
    Commented May 29, 2014 at 10:29

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