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We study the maximum likelihood (ML) degree of linear concentration models inalgebraic statistics. We relate it to an intersection problem on the variety ofcomplete quadrics. This allows us to provide an explicit, basic, albeit of highcomputational complexity, formula for the ML-degree. The variety of completequadrics is an exact analog for symmetric matrices of the permutohedron varietyfor the diagonal matrices.

Maximum likelihood degree, complete quadrics and ${\mathbb C}^*$-action