International Congress of Mathematicians

A P • • = M E1[ - P ° ) C CmAsymptotics of Polynomials and Eigenfunctions 7372.4. Polynomials with fixed Newton polytopeThe well-known Bernstein-Kouchnirenko theorem states that the number ofsimultaneous zeros of (a generic family of) m polynomials with Newton polytopeP equals m\Vol(P). Recall that the Newton polytope Pf of a polynomial is theconvex hull of its support Sf = {a £ Z TO : c a ^ 0}. Using the homogenization mapf —¥ F, the space of polynomials / whose Newton polytope Pf contained in P maybeidentified with a subspaceH 0 (CV m ,O(p),P) = {F£ H 0 (CV m ,O(pj) :P f CP} (2.4)ofH 0 (CV m ,O(pj).The problem we address in this section is:• Problem 3 How does the Newton polytope influence on the distribution ofzeros of polynomials?Again, one could ask the same question about L 2 mass, critical points and soon and obtain a similar story. In [19] we explore this influence in a statistical andasymptotic sense. The main theme is that for each property of polynomials understudy, P gives rise to classically allowed regions where the behavior is the same asif no condition were placed on the polynomials, and classically forbidden regionswhere the behavior is exotic.Let us define these terms. If P C R is a convex integral polytope, then theclassically allowed region for polynomials in H°(CW m ,ö(p),P) is the set.-i'' 1(where P° denotes the interior of P), and the classically forbidden region is its(II 2 i i 2 \-,finl n 2, • • •, -IYTJ[|2 I is the moment map ofCP.The result alluded to above is statistical. Since we view the polytope P ofdegree p as placing a condition on the Gaussian ensemble of SU(P) polynomialsof degree p, we endow H°(CW m ,ö(p),P) with the conditional probability measure1S\Pdj6\p(s)= -^e-^dX, «=^A Q^, (2.5)a£P " "where the coefficients A Q are again independent complex Gaussian random variableswith mean zero and variance one.Our simplest result concerns the the expected density E|p(Z/ li ... i / m ) of thesimultaneous zeros of (fi, • • •, f m ) chosen independently from H°(CW m ,ö(p),P).It is the measure on C* TO given byV\p(Z fu ..., f J(U)d%\p(fi)--- I'd % \ P (f m ) [#{z £ U : fi(z) = • • • = f m (z) = 0}] , (2.6)