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For the chromatic polynomials of graphs we have two nice theorems which describe the behavior of their zeros: Thomassen proved that the set of real zeros of all chromatic polynomials is the union of $\lbrace 0,1\rbrace$ and a set of points which is dense in $(32/27,\infty )$. Sokal proved that the corresponding set of complex zeros is dense in $\mathbb C$.

I am interested whether there are analogous results for Tutte polynomials and their zeros. In particular, is it known whether the union of all curves in $\mathbb C^2$ cut out by the Tutte polynomials of graphs is dense in $\mathbb C^2$? Pointers towards the behavior of real zeros are also welcome!

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Interesting question that seems hard. The union of curves can't be dense in $\mathbb{R}^2$, because no Tutte polynomial of any graph (or for that matter, matroid) can have a root in the open first quadrant. But that argument doesn't work for $\mathbb{C}$. Do you know if $(2,2)$ is in the closure of the union of Tutte curves? If so, it would be interesting to see how. I'd also be curious about what happens if you replace "graph" with "matroid". –  Jeremy Martin Jul 31 '12 at 4:43
    
Dear @F. C., I am leaving a comment here because you made six edits to old questions within a period of less than an hour a couple of days ago. Please be aware that you should not edit more than three old questions each day, as they get bumped to the front page. Please read the discussion at the following meta thread for more information: meta.mathoverflow.net/questions/599/… –  Ricardo Andrade Feb 6 at 23:37

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