HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

Definitions

Definition (stability)

A polynomial q is said to be stable if q(z) 6= 0 whenever eachcoordinate zj is in the strict upper half plane.

Definition (hyperbolicity)

A homogeneous polynomial p of degree m is said to be hyperbolicin direction x 2 Rd if p(x+ iy) 6= 0 for all y 2 Rd.

Proposition (equivalences)

Hyperbolicity in direction x is equivalent to the univariatepolynomial p(y + tx) having only real roots for all y 2 Rd.

A real homogeneous polynomial is stable if and only if it ishyperbolic in every direction in the positive orthant.

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

Cones of hyperbolicity

The real variety V := {p = 0} ✓ Rd plays a special role inhyperbolicity theory.

Proposition (cones of hyperbolicity)

Let ⇠ be a direction of hperbolicity for p and let K(p, ⇠) denote theconnectied component of the set Rd \ V that contains ⇠.

p is hyperbolic in direction x for every x 2 K(p, ⇠).

The set K(p, ⇠) is an open convex cone; we call this a coneof hyperbolicity for p.

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

Stable evolution of PDE’s

Let p be a polynomial in d variables and denote by Dp theoperator p(@/@x) obtained by replacing each xi by @/@xi.

Let r be a vector in Rd, let Hr be the hyperplane orthogonal to r,and consider the equation

Dp(f) = 0 (1)

in the halfspace {r · x � 0} with boundary conditions specified onHr (typically, f and its first d� 1 normal derivatives).

We say that (1) evolves stably in direction r if convergence of theboundary conditions to 0 implies convergence1 of the solution to 0.

1Uniform convergence of the function and its derivatives on compact sets.

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

Garding’s Theorem

Theorem ([Gar51, Theorem III])

The equation Dpf = 0 evolves stably in direction r if and only if pis hyperbolic in direction r .

Let us see why this should be true.

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

Inequalities that make this work well

Suppose � 2 C 3 is defined on the interior of a convex cone K , ishomogeneous of degree �m, and goes to infinity at @K .

If F := log � satisfies

|D3F (x)[h, h, h]| 2�D2F (x)[x , x ]

�3/2

then � serves as a barrier function. If, in addition,

|DF (x)[h]|2 mD2F (x)[h, h]|

then the interior method will converge well.

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

van der Waerden conjecture

Theorem (van der Waerden conjecture)

The minimum permanent of an n ⇥ n doubly stochastic matrix isn!/nn, uniquely obtained when all entries are 1/n.

This was proved by Falikman (value of the minimum) andEgorychev (uniqueness) in 1981.

Gurvits (2008) found a much simpler proof using stability. Let Cnbe the set of homogeneous polynomials of degree n in n variableswith nonnegative coe�cients.

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

Permanent as a stable polynomial in Cn

If A is an n⇥ n matrix then its permanent may be represented as amixed partial derivative of a homogeneous stable polynoimal.

Define

pA(x) :=nY

i=1

nX

j=1

aijxj .

Proposition (Gurvits 2008)

The polynomial p is stable and

per (A) =@n

@x1 · · · @xnpA(0, . . . , 0) .

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

References I

P. Branden, J. Haglund, M. Visontai, and D. Wagner.Proof of the monotone column permanent conjecture.In FPSAC 2009, volume AK, pages 443–454, Nancy, 2009. Assoc. DiscreteMath. Theor. Comput. Sci.

Y. Baryshnikov and R. Pemantle.Asymptotics of multivariate sequences, part III: quadratic points.Adv. Math., 228:3127–3206, 2011.

G. Egorychev.The solution of van der Waerden’s problem for permanents.Adv. Math., 42:299–305, 1981.

D. Falikman.Proof of the van der Waerden’s conjecture on the permanent of a doublystochastic matrix.Mat. Zametki, 29:931–938, 1981.(in Russian).

HyperbolicityPartial di↵erential equations

Other uses: interior point methods for convex programmingOther uses: matrix theorems and conjectures

References II

L. Garding.Linear hyperbolic partial di↵erential equations with constant coe�cients.Acta Math., 85:1–62, 1951.

O. Guler.Hyperbolic polynomials and interior point methods for convex programming.Math. Oper. Res., 22:350–377, 1997.

L. Gurvits.The van der Waerden conjecture for mixed discriminants.Adv. Math., 200:435–454, 2006.

J. Haglund, K. Ono, and D. G. Wagner.Theorems and conjectures involving rook polynomials with only real zeros.In Topics in Number Theory, volume 467 of Math. Appl., pages 207–221. KluwerAcademic Publishers, Dordrecht, 1999.

R. Pemantle.Hyperbolicity and stable polynomials in combinatorics and probability.In Current Developments in Mathematics, pages 57–124. International Press,Somerville, MA, 2012.