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The main purpose of this note is to show how Sturm-Habicht Sequence can be generalized to the multivariate case and used to compute the number of real solutions of a polynomial system of equations with a finite number of complex solutions. Using the same techniques, some formulae counting the number of real solutions of such polynomial systems of equations inside n-dimensional rectangles or triangles in the plane are presented.

Mutating seeds: types $𝔸$ and $\tilde{𝔸}$ .

Ibrahim Assem, Christophe Reutenauer (2012)

Annales mathématiques Blaise Pascal

In the cases $𝔸$ and $\tilde{𝔸}$ , we describe the seeds obtained by sequences of mutations from an initial seed. In the $\tilde{𝔸}$ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.

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