  We give a sharp lower bound on the capacity of a real stable polynomial,
depending only on the value of its gradient at $x = 1$. This result implies a
sharp improvement to a similar inequality proved by
Linial-Samorodnitsky-Wigderson in 2000, which was crucial to the analysis of
their permanent approximation algorithm. Such inequalities have played an
important role in the recent work on operator scaling and its generalizations
and applications, and in fact we use our bound to construct a new scaling
algorithm for real stable polynomials.
  In addition, we give a strong improvement on previous lower bounds of the
capacity of a non-homogeneous real stable polynomial, depending only on the
value of its gradient at $x = 1$. Crucially, this new bound is independent of
the degree of the polynomial, and has singly exponential dependence on the
number of variables. This compares favorably to the bounds used recently in the
fantastic work of Karlin-Klein-Oveis Gharan to give an improved approximation
factor for metric TSP, where this dependence is doubly exponential. Such bounds
were conjectured to exist by the authors, and thus our new bound should imply
further improvement to the approximation factor for metric TSP.
  The new technique we develop to prove this bound is productization, which
says that any real stable polynomial can be approximated at any point in the
positive orthant by a product of linear forms. Beyond the results of this
paper, our main hope is that this new technique will allow us to avoid
"frightening technicalities", in the words of Laurent and Schrijver, that often
accompany combinatorial lower bounds.
