In the introduction to their book "Discriminants, resultants and multidimensional determinant", the authors state a very intriguing observation concerning the coefficients of monomials appearing in $A$-discriminants, which are of the form $vol(\sigma_i)^{vol(\sigma_i)}$, for $\sigma_i$ a simplex (page 5):

"The expression $\prod_ivol(\sigma_i)^{vol(\sigma_i)}$ (or, rather, its logarithm $\sum_i vol(\sigma_i) \log (vol(\sigma_i))$) brings to mind the entropy of a probability distribution. It would be interesting to find a "probabilistic" reason for its appearance in discriminants."

Does any know about the status of this observation, or about references discussing it?

  • $\begingroup$ For those of us intrigued by your question but without the book to hand, could you explain a bit more of the context? Thanks. $\endgroup$ Mar 4, 2014 at 11:29
  • $\begingroup$ There's now a link to the mentioned page, "courtesy" of Google Books. $\endgroup$ Mar 4, 2014 at 12:45
  • $\begingroup$ Thanks. The link didn't work for me ("you've reached your page viewing limit" or some such), but when I searched for it myself, I found it. Google, eh. $\endgroup$ Mar 4, 2014 at 13:45
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    $\begingroup$ The following certainly isn't an answer to your question, but I thought a bit about some related things a while ago; maybe there's some connection between discriminants and entropy via geometric invariants such as mixed volume. See here, especially Mark Meckes's pointer to Richard Gardner's article. $\endgroup$ Mar 4, 2014 at 13:48
  • $\begingroup$ That's a nice writing on mixed volume. I didn't know it so resembled the determinant. $\endgroup$ Mar 5, 2014 at 11:11


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