Counting Matchings via Capacity Preserving Operators
aa r X i v : . [ m a t h . C O ] A ug Counting Matchings via Capacity Preserving Operators
Jonathan LeakeLeonid GurvitsAugust 14, 2019
Abstract
The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to givedrastically simplified proofs of the Van der Waerden lower bound for permanents of doubly stochasticmatrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminalwork, the notion of capacity has been utilized to bound various combinatorial quantities and to givepolynomial-time algorithms to approximate such quantities (e.g., the number of bases of a matroid).These types of results are often proven by giving bounds on how much a particular differential operatorcan change the capacity of a given polynomial. In this paper, we unify the theory surrounding suchcapacity preserving operators by giving tight capacity preservation bounds for all nondegenerate realstability preservers. We then use this theory to give a new proof of a recent result of Csikv´ari, whichsettled Friedland’s lower matching conjecture.
Definition 1.1.
A polynomial p ∈ R [ x , ..., x n ] is said to be real stable if p does not vanish when all inputslie in the complex upper half-plane.Over the past few decades, the theory of real stability has found various applications, particularly withincombinatorics, probability, computer science, and optimization (e.g., see [BB09b], [COSW04] and referencestherein). Classic examples include the multivariate matching polynomial and the spanning tree polynomial,both of which are real stable for any given graph. The role that polynomials often play in these applicationsis that of conceptual unification: various natural operations that one may apply to a given type of object canoften be represented as natural operations applied to associated polynomials. For the matching polynomialdeletion and contraction correspond to certain derivatives, and for the spanning tree polynomial this ideaextends to the minors of a matroid in general. Even in optimization (specifically hyperbolic programming),certain relaxations of convex domains translate into directional derivatives of associated polynomials in asimilar way [Ren06].Real stability then adds extra information that may be useful to track. For example, the real stability ofthe matching polynomial easily implies that the number of size k matchings of a graph forms a log-concavesequence [HL72]. As it turns out, real stability is far more generally connected to log-concavity than this,and we will see this at play in the main results of this paper. Specifically, the so-called strong Rayleighinequalities (e.g., see [Br¨a07]) will play a crucial role in our analysis. Related inequalities have recently havegained importance through the exciting work on a so-called Hodge theory for matroids ([AHK18]). Resultssimilar to those discussed here can even be extended to basis generating polynomials of matroids in general(not all of which are real stable) (see [HSW18], [AGV18]).The particular line into which this paper falls then begins with the work of the second author, who ina series of papers (e.g., see [Gur08]) gave a vast generalization of the Van der Waerden lower bound forpermanents of doubly stochastic matrices and the Schrijver lower bound on the number of perfect matchingsof regular graphs. In particular, he showed that a related inequality holds for real stable polynomials ingeneral, and then derives each of the referenced results as corollaries. His inequality describes how much the1erivative can affect a particular analytic quantity called the capacity of a polynomial, and we now state itformally. Throughout this paper we use the notation x α := Q k x α k k . Definition 1.2 (Gurvits) . Given a polynomial p ∈ R [ x , ..., x n ] with non-negative coefficients and a vector α ∈ R n with non-negative entries, we define the α -capacity of p as:Cap α ( p ) := inf x> p ( x ) x α Theorem 3.1 (Gurvits) . Let p ∈ R [ x , ..., x n ] be a real stable polynomial of degree at most λ k in x k withnon-negative coefficients. Then: Cap (1 n − ) (cid:0) ∂ x k p | x k =0 (cid:1) Cap (1 n ) ( p ) ≥ (cid:18) λ k − λ k (cid:19) λ k − Here, (1 j ) denotes the all-ones vector of length j . The way one should interpret this result is as a statement about the capacity preservation properties ofthe derivative. That is, taking a partial derivative of a real stable polynomial (and then evaluating to 0) canonly decrease the capacity of that polynomial by at most the stated multiplicative factor.For those familiar with the real stability literature, the concept of preservation properties of a linearoperator (specifically that of the derivative here) is not new. Perhaps the most essential result in the theoryis the Borcea-Br¨and´en characterization [BB09a], which characterizes all linear operators on polynomialswhich preserve the property of being real stable. This result relies on the concept of the symbol of a linearoperator T , denoted Symb( T ), which is a single specific polynomial (or power series) associated to T . Thatsaid, we give the gist of the characterization here but delay the definition of the symbol and the formalstatement of the theorem until later. Theorem 2.5/2.8 (Borcea-Br¨and´en) . Let T be a real linear operator on polynomials. Then morally speaking, T preserves the property of being real stable if and only if Symb( T ) is real stable. This theorem says that the symbol of a linear operator T holds the real stability preservation informationof T . In this paper, we make use of this concept by showing that the symbol also holds the capacitypreservation information of T . That is, we combine the ideas of Gurvits and of Borcea and Br¨and´en tocreate a theory of capacity preserving operators. Our main results in this direction are stated as follows. Itis important to note that Symb will take on two different definitions in the formal statements of the followingtheorems (see Definitions 2.4 and 2.7), and we will explicate this rigorously later. (For those unfamiliar, theLaguerre-P´olya class consists of limits of real stable polynomials.) Theorem 4.11 (Bounded degree) . Let T be a linear operator on polynomials of degree at most λ k in x k ,such that Symb( T ) is real stable with non-negative coefficients. Further, let p ∈ R [ x , ..., x n ] be a real stablepolynomial of degree at most λ k in x k with non-negative coefficients. Then for any sensible non-negativevectors α, β ∈ R n : Cap β ( T ( p ))Cap α ( p ) ≥ α α ( λ − α ) λ − α λ λ Cap ( α,β ) (Symb( T )) Further, this bound is tight for fixed
T, α, β . Theorem 4.12 (Unbounded degree) . Let T be a linear operator on polynomials of any degree, such that Symb( T ) is in the Laguerre-P´olya class with non-negative coefficients. Further, let p ∈ R [ x , ..., x n ] be anyreal stable polynomial with non-negative coefficients. Then for any sensible non-negative vectors α, β ∈ R n : Cap β ( T ( p ))Cap α ( p ) ≥ e − α α α Cap ( α,β ) (Symb( T )) Further, this bound is tight for fixed
T, α, β . T = ∂ x k | x k =0 .Our main application is then related to counting matchings of regular bipartite graphs. Counting thenumber of matchings in a graph is related to the monomer-dimer problem of evaluating/approximating themonomer-dimer partition function of a given graph. This problem is one of the oldest and most importantproblems in statistical physics, with much of the importance being due to the famous paper of Heilmann andLieb [HL72]. Their results on the location of phase transitions of the partition function (i.e., the locationof zeros of the matching polynomial) have had widespread influence, even playing a crucial role in the(somewhat) recent resolution of Kadison-Singer by Marcus, Spielman, and Srivastava [MSS15b].Specifically, we give a simpler proof of Csikv´ari’s bound on the number of k -matchings of a biregularbipartite graph [Csi14]. This result generalizes Schrijver’s inequality and is actually a strengthening ofFriedland’s lower matching conjecture (see [FKM08]). The computations involved in this new proof neverexceed the level of basic calculus. This was one of the most remarkable features of Gurvits’ original result,and this theme continues to play out here. We state Csikv´ari’s result now. Theorem 3.3 (Csikv´ari) . Let G be an ( a, b ) -biregular bipartite graph with ( m, n ) -bipartitioned vertices (sothat am = bn is the number of edges of G ). Then the number of size- k matchings of G is bounded as follows: µ k ( G ) ≥ (cid:18) nk (cid:19) ( ab ) k m m ( ma − k ) ma − k ( ma ) ma ( m − k ) m − k We also note that partial results toward such a bound, using techniques similar to those used in thispaper, were achieved prior to Csikv´ari’s result. First in [FKM08], the original lower matching conjecture(for regular graphs) was settled for degree 2 and for k ≤
4. Further in [FG08], partial results are given forthe asymptotic version of the lower matching conjecture, and in [Gur11] this asymptotic version is settled.In these last two papers, stable polynomials and results derived from Theorem 3.1 were used.Beyond these specific applications, one of the main purposes of this paper is to unify the various resultsthat fit into the lineage of the concept of capacity. Some of these are inequalities for specific combinatorialquantities ([Gur08], [Gur09], [Gur11]), some are approximation algorithms for those quantities ([AGV18],[SV17]), and some are capacity preservation results similar to those in this paper (particularly [AG17]).The rest of this paper is outlined as follows. In §
2, we discuss some preliminary facts about real stabilityand capacity. In §
3, we discuss applications of the capacity preservation theory. In §
4, we prove the maininequalities. In §
5, we discuss some continuity properties of capacity.
Acknowledgements
The first author is thankful to Peter Csikv´ari for discussions on his bound and on a number of relatedtopics and results. The first author is also thankful to Nick Ryder for discussions on topics related to thepolynomial theory used in this paper.
We first discuss some basics of the theories of real stability and of capacity. This section will consist mainly ofwell-known and/or standard results that will enable us to state our main results in the next section formally.Other results needed to prove the main theorems will be left to later sections.
Let C , R , Z , N denote the complex numbers, real numbers, integers, and positive integers respectively. Also,let R + and R ++ denote the non-negative and positive reals respectively, and let Z + denote the non-negative3ntegers. With this we let K [ x , ..., x n ] denote the set of polynomials with coefficients in K , where K can beany of the previously defined sets of numbers. Further, for λ ∈ Z n + we let K λ [ x , ..., x n ] denote the set ofpolynomials of degree at most λ k in x k with coefficients in K .For µ, λ ∈ Z n + , we define µ ! := Q k ( µ k !) and (cid:0) λµ (cid:1) := λ ! µ !( λ − µ )! . For x, α ∈ R n + we define α ≤ x via α k ≤ x k for all k , we define xα := Q k x k α k , and we define x α := Q k x α k k as in the definition of capacity. We also let(1 n ) ∈ Z n + denote the all-ones vector of length n . Finally for p ∈ K [ x , ..., x n ] we let p µ denote the coefficientof p corresponding to the term x µ , and in this vein we will sometimes let x refer to the vector of variables( x , ..., x n ). We call a polynomial p ∈ R [ x , ..., x n ] real stable if p ≡ p ( x , ..., x n ) = 0 whenever all x k are in the upperhalf-plane. Note that for n = 1, p is a univariate polynomial and real stability is equivalent to having allreal roots. The theory of stable polynomials enjoys a nice inductive structure deriving from a large class oflinear operators on polynomials which preserve the property of being real stable. We called such operators real stability preservers , and the most basic of these are given as follows. Proposition 2.1 (Basic real stability preservers) . Let p ∈ R λ [ x , ..., x n ] be real stable. Then the followingare also real stable.1. Permutation: p ( x σ (1) , ..., x σ ( n ) ) for any σ ∈ S n
2. Scaling: p ( a x , ..., a n x n ) for any fixed a ∈ R n +
3. Specialization: p ( b, x , ..., x n ) for any fixed b ∈ R
4. Inversion: x λ p ( x − , ..., x − n )
5. Differentiation: ∂ x k p for any k
6. Diagonalization: p | x j = x k for any j, k A classical but more interesting real stability preserver is polarization . Polarization plays a crucial role inthe theory of real stability preservers, as it allows one to restrict to polynomials of degree at most 1 in everyvariable. We will see later that polarization also plays a crucial role in the theory of capacity preservers.
Definition 2.2.
Given q ∈ R d [ x ], we define Pol d ( q ) to be the unique symmetric f ∈ R (1 d ) [ x , ..., x d ] suchthat f ( x, ..., x ) = q ( x ). Given p ∈ R λ [ x , ..., x n ], we define Pol λ ( p ) := (Pol λ ◦ · · · ◦ Pol λ n )( p ), where Pol λ k acts on the variable x k for each k . Note that Pol λ ( p ) ∈ R (1 λ ··· + λn ) [ x , , ..., x n,λ n ]. Proposition 2.3 ([Wal22]) . Given p ∈ R λ [ x , ..., x n ] , we have that p is real-stable iff Pol λ ( p ) is real stable. Beyond these basic real stability preservers, various preservation results regarding different classes ofoperators have been proven over the past century or so. In 2008 many of these results were encapsulatedand vastly generalized in the Borcea-Br¨and´en characterization, which gives a useful equivalent condition fora linear operator to be a real stability preserver. We mentioned this result in the introduction, and now wepresent it formally. To that end, we first define the symbol of an operator , a crucial concept to the rest ofthis paper.
Definition 2.4 (Bounded-degree symbol) . Let T : R λ [ x , ..., x n ] → R γ [ x , ..., x m ] be a linear operator onpolynomials. We define Symb λ ( T ) ∈ R ( λ,γ ) [ z , ..., z n , x , ..., x m ] as follows, where T acts only on the x variables. Symb λ ( T ) := T [(1 + xz ) λ ] = X ≤ µ ≤ λ (cid:18) λµ (cid:19) z µ T ( x µ )We may simply write Symb( T ) when λ is clear from the context. (Note that this definition is slightly differentfrom that of [BB09a], but this difference is inconsequential.)4he characterization then essentially says that T preserves real stability if and only if Symb λ ( T ) is realstable, with the exception of a certain degeneracy case. We state the full statement of the characterization,for bounded-degree operators. Theorem 2.5 (Borcea-Br¨and´en) . Let T : R λ [ x , ..., x n ] → R γ [ x , ..., x m ] be a linear operator on polynomials.Then T preserves real stability if and only if one of the following holds.1. Symb λ ( T ) is real stable.2. Symb λ ( T )( z , ..., z n , − x , ..., − x m ) is real stable.3. The image of T is of degree at most 2 and consists only of real stable polynomials. Notice that the above definition and result deal only with operators which only allow inputs up to acertain fixed degree. And this is important to note, as the symbol changes based upon which degree isbeing considered. For operators which do not inherently depend on some fixed maximum degree (e.g., thederivative), there is another symbol definition and characterization result.Of course, the degree of the symbol above is the same as the maximum degree of the input polynomials.So if one were to define a symbol for operators with no bound on the input degree, it is likely that the symbolwould not have a bound on its degree. This is where the
Laguerre-P´olya class comes in. This is a class ofentire functions in C n , defined as follows. Definition 2.6.
A function f is said to be in the LP (Laguerre-P´olya) class in the variables x , ..., x n , if f isthe limit (uniformly on compact sets) of real stable polynomials in R [ x , ..., x n ]. If f is the limit of real stablepolynomials in R + [ x , ..., x n ], then we say f is in the LP + class. In these cases, we write f ∈ LP [ x , ..., x n ]and f ∈ LP + [ x , ..., x n ] respectively.There are interesting equivalent definitions for these classes of functions (e.g., see [CC89]), but we omitthem here. With this class of functions we can state the Borcea-Br¨and´en characterization for operators withno dependence on the degree of the input polynomial. First though we need to define the “transcendental”symbol. Definition 2.7 (Transcendental symbol) . Let T : R [ x , ..., x n ] → R [ x , ..., x m ] be a linear operator onpolynomials. We define Symb ∞ ( T ) as a formal power series in z , ...., z n (with polynomial coefficients in x , ..., x m ) as follows, where T acts only on the x variables.Symb ∞ ( T ) := T [ e x · z ] = X ≤ µ µ ! z µ T ( x µ ) Theorem 2.8 (Borcea-Br¨and´en) . Let T : R [ x , ..., x n ] → R [ x , ..., x m ] be a linear operator on polynomials.Then T preserves real stability if and only if one of the following holds.1. Symb ∞ ( T ) ∈ LP [ z , ...., z n , x , ..., x m ] Symb ∞ ( T )( z , ..., z n , − x , ..., − x m ) ∈ LP [ z , ...., z n , x , ..., x m ]
3. The image of T is of degree at most 2 and consists only of real stable polynomials. Recall the definition of capacity: Cap α ( p ) := inf x> p ( x ) x α In general, the conceptual meaning of capacity is not completely understood. However, in this sectionwe hope to illuminate some of its basic features. This will include its connections to the coefficients of a5olynomial, to probabilistic interpretations of polynomials, to the AM-GM inequality, and to the Legendre(Fenchel) transformation.As discussed in the introduction, the sort of capacity results we will be interested in are those of capacitypreservation (that is, bounds on how much the capacity can change under various operations). In fact, ouruse of the Borcea-Br¨and´en characterization consists in combining it with capacity bounds in order to givesomething like a characterization of capacity preservers. This can be seen as an analytic refinement of thecharacterization: not only do such operators preserve stability, but they also preserve capacity. That said,we now state a few basic properties and interpretations of capacity that will be needed to state and discussthis analytic refinement. First recall the definitions of the Newton polytope and the support of a polynomial.
Definition 2.9.
Given p ∈ R [ x , ..., x n ], the Newton polytope of p , denoted Newt( p ), is the convex hull ofthe support of p . The support of p , denoted supp( p ), is the set of all µ ∈ Z n + such that x µ has a non-zerocoefficient in p .Capacity is perhaps most basically understood as a quantity which mediates between the coefficients of p and the evaluations of p . For example, if µ ∈ supp( p ) then: p µ ≤ Cap µ ( p ) ≤ p (1 , ..., p ∈ R (1 n )+ [ x , ..., x n ] and p (1 , ...,
1) = 1, then p can beconsidered as the probability generating function for some discrete distribution on supp( p ). In this case, asimple proof demonstrates: Fact 2.10.
Let p ∈ R (1 n )+ [ x , ..., x n ] be the probability generating function for some distribution ν . Then:1. ≤ Cap α ( p ) ≤ for all α ∈ R n + .2. Cap α ( p ) = 1 if and only if α is the vector of marginal probabilities of ν .Proof. (1) is straightforward, and (2) follows from concavity of log (e.g., see [Gur08], Fact 2.2) and the factthat Cap α ( p ) = 1 implies p ( x ) x α is minimized at the all-ones vector.The following “log-exponential polynomial” associated to p has some nice properties which often makesit convenient to use in the context of capacity. These properties also shed light on the potential connectionbetween capacity, convexity, and the Legendre transformation (consider the expressions which show up inFact 2.12 below). Definition 2.11.
Given a polynomial p ∈ R + [ x , ..., x n ], we let capitalized P denote the following function: P ( x ) := log( p (exp( x ))) = log X µ p µ e µ · x Fact 2.12.
Given p ∈ R + [ x , ..., x n ] , consider P as defined above. We have:1. Cap α ( p ) = exp inf x ∈ R n ( P ( x ) − α · x ) P ( x ) − α · x is convex in R n for any α ∈ R n . The next result is essentially a corollary of the AM-GM inequality. In a sense, this inequality is thefoundational result that makes the notion of capacity so useful. Because of this we provide a partial proofof the following result, taken from [AG17].
Fact 2.13.
For p ∈ R + [ x , ..., x n ] , P defined as above, and α ∈ R n + , the following are equivalent.1. α ∈ Newt( p ) Cap α ( p ) > . P ( x ) − α · x is bounded below.Proof. That (2) ⇔ (3) follows from the previous fact. We now prove (1) ⇒ (2). The (2) ⇒ (1) direction alsohas a short proof, based on a separating hyperplane for α and Newt( p ) whenever α Newt( p ). The detailscan be found in Fact 2.18 of [AG17].Suppose that α ∈ Newt( p ). So, α = P µ ∈ S c µ µ , where S ⊂ supp( p ), c µ >
0, and P µ ∈ S c µ = 1. Using theAM-GM inequality and the fact that the coefficients of p are non-negative, we have the following for x ∈ R n + : p ( x ) ≥ X µ ∈ S p µ x µ = X µ ∈ S c µ p µ x µ c µ ≥ Y µ ∈ S (cid:18) p µ x µ c µ (cid:19) c µ = x α Y µ ∈ S (cid:18) p µ c µ (cid:19) c µ This then implies: Cap α ( p ) = inf x> p ( x ) x α ≥ Y µ ∈ S (cid:18) p µ c µ (cid:19) c µ > α which are in the Newton polytope ofthe relevant polynomials. Other α can be considered but most results will then become trivial. That said,we will often make this assumption about α without explicitly stating it.The next result emulates Proposition 2.1 (the basic real stability preservers) by giving a collection ofbasic capacity preserving operators. Note that these results are either equalities, or give something of theform Cap( T ( p )) ≥ c T · Cap( p ) for various operators T . Proposition 2.14 (Basic capacity preservers) . For p, q ∈ R + λ [ x , ..., x n ] and α, β ∈ R n + , we have:1. Scaling: Cap α ( bp ) = b · Cap α ( p ) for b ∈ R +
2. Product:
Cap α + β ( pq ) ≥ Cap α ( p ) Cap β ( q )
3. Disjoint product:
Cap ( α,β ) ( p ( x ) q ( z )) = Cap α ( p ) Cap β ( q )
4. Evaluation:
Cap ( α ,...,α n − ) ( p ( x , ..., x n − , y n )) ≥ y α n n Cap α ( p ) for y n ∈ R +
5. External field:
Cap α ( p ( cx )) = c α Cap α ( p ) for c ∈ R n +
6. Inversion:
Cap ( λ − α ) ( x λ p ( x − , ..., x − n )) = Cap α ( p )
7. Concavity:
Cap α ( bp + cq ) ≥ b · Cap α ( p ) + c · Cap α ( q ) for b, c ∈ R +
8. Diagonalization:
Cap P α k ( p ( x, ..., x )) ≥ Cap α ( p )
9. Symmetric diagonalization:
Cap n · α ( p ( x, ..., x )) = Cap α ( p ) if α = ( α , ..., α ) and p is symmetric10. Homogenization: Cap ( α,λ − α ) (Hmg λ ( p )) = Cap α ( p ) Proof.
Symmetric diagonalization is the only nontrivial property, and it is a consequence of the AM-GMinequality. First of all, we automatically have (the diagonalization inequality):Cap n · α ( p ( x, ..., x )) = inf x> p ( x, ..., x ) x α · · · x α ≥ inf x> p ( x , ..., x n ) x α · · · x α n = Cap α ( p )For the other direction, fix x ∈ R n + and let y := ( x · · · x n ) /n . Further, let S ( p ) denote the symmetrizationof p . For any µ ∈ Z n + , the AM-GM inequality gives: S ( x µ ) = 1 n ! X σ ∈ S n x µ σ (1) · · · x µ n σ ( n ) ≥ Y σ ∈ S n x µ σ (1) · · · x µ n σ ( n ) ! /n ! = Y j,k x µ k j /n = y µ · · · y µ n x α = x α · · · x α n = y n · α . Since p is symmetric, we then have the following: p ( x ) x α = S ( p )( x ) x α = X µ ∈ supp( p ) p µ S ( x µ ) x α ≥ X µ ∈ supp( p ) p µ y µ · · · y µ n y n · α = p ( y, ..., y ) y n · α That is, for any x ∈ R n + , there is a y ∈ R + such that p ( x ) x α ≥ p ( y,...,y ) y n · α . Therefore:Cap α ( p ) ≥ Cap n · α ( p ( x, ..., x ))This completes the proof.Many of these operations are similar to those that preserve real stability. This is to be expected, as wehope to combine the two theories. In this vein, we now discuss the capacity preservation properties of thepolarization operator. As it does for real stability preservers, polarization will play a crucial role in workingout the theory of capacity preservers. To state this result, we define the polarization of the vector α asfollows, where each value α k λ k shows up λ k times:Pol λ ( α ) := (cid:18) α λ , ..., α λ , α λ , ..., α λ , ..., α n λ n , ..., α n λ n (cid:19) Proposition 2.15.
Given p ∈ R λ + [ x , ..., x n ] and α ∈ R n + , we have that Cap
Pol λ ( α ) (Pol λ ( p )) = Cap α ( p ) .Proof. We essentially apply the diagonalization property to each variable in succession. Specifically, we have:Cap α ( p ) = inf y ,...,y n − > y α · · · y α n − n − inf x n > p ( y , ..., y n − , x n ) x α n n = inf y ,...,y n − > y α · · · y α n − n − Cap α n ( p ( y , ..., y n − , x n ))= inf y ,...,y n − > y α · · · y α n − n − Cap
Pol λn ( α n ) (Pol λ n ( p ( y , ..., y n − , · )))By now rearranging the inf’s in the last expression above, we can let inf y n − > be the inner-most inf. Wecan then apply the above argument again, and this will work for every y k in succession. At the end of thisprocess, we obtain:Cap α ( p ) = Cap (Pol λ ( α ) ,..., Pol λn ( α n )) (Pol λ ◦ · · · ◦ Pol λ n ( p )) = Cap Pol λ ( α ) (Pol λ ( p ))Note that the two main results on polarization—capacity preservation and real stability preservation—imply that we only really need to prove our results in the multiaffine case (i.e., where polynomials are ofdegree at most 1 in each variable). We will make use of this reduction when we prove our technical resultsin § Lemma 2.16.
For c, α ∈ R n + and m := P k α k , we have the following: Cap α (( c · x ) m ) ≡ Cap α (cid:18)(cid:0) X k c k x k (cid:1) m (cid:19) = (cid:16) mcα (cid:17) α roof. Note first that: Cap α (( c · x ) m ) = (cid:16) Cap αm ( c · x ) (cid:17) m To compute Cap αm ( c · x ), we use calculus. Let β := αm , and for now we assume that β > c > ∂ x k (cid:16) c · xx β (cid:17) = x β c k − β k x β − δ k ( c · x ) x β = x k c k − β k ( c · x ) x β + δ k That is, the gradient of c · xx β is the 0 vector whenever c k β k x k = c · x for all k . And in fact, any vector satisfyingthose conditions should minimize c · xx β , by homogeneity. Since P k β k = 1, the vector x k := β k c k satisfies theconditions. This implies: Cap β ( c · x ) = c · ( β/c )( β/c ) β = (cid:18) cβ (cid:19) β Therefore: Cap α (( c · x ) m ) = (cid:18) cβ (cid:19) mβ = (cid:16) mcα (cid:17) α We now formally state and discuss our main results and their applications. As mentioned above, we willemulate the Borcea-Br¨and´en characterization for capacity preservers. Further, we will also demonstrate howour results encapsulate many of the previous results regarding capacity. With this in mind, we first giveour main capacity preservation results: one for bounded degree operators a la Theorem 2.5, and one forunbounded degree operators a la Theorem 2.8. Notice that the unbounded degree case is something like alimit of the bounded degree case: the scalar α α ( λ − α ) λ − α λ λ is approximately (cid:0) αλ (cid:1) α e − α as λ → ∞ . (The proofof Theorem 4.8 shows why the extra λ − α factor disappears.) Theorem 4.11 (Bounded degree) . Let T : R λ + [ x , ..., x n ] → R γ + [ x , ..., x m ] be a linear operator with realstable symbol. Then for any α ∈ R n + , any β ∈ R m + , and any real stable p ∈ R λ + [ x , ..., x n ] we have: Cap β ( T ( p ))Cap α ( p ) ≥ α α ( λ − α ) λ − α λ λ Cap ( α,β ) (Symb λ ( T )) Further, this bound is tight for fixed T , α , and β . Theorem 4.12 (Unbounded degree) . Let T : R + [ x , ..., x n ] → R + [ x , ..., x m ] be a linear operator with realstable symbol. Then for any α ∈ R n + , any β ∈ R m + , and any real stable p ∈ R + [ x , ..., x n ] we have: Cap β ( T ( p ))Cap α ( p ) ≥ α α e − α Cap ( α,β ) (Symb ∞ ( T )) Further, this bound is tight for fixed T , α , and β . Note that by Theorems 2.5 and 2.8, the above theorems apply to real stability preservers of rank greaterthan 2 (see Corollaries 4.13 and 4.14).
With these results in hand, we now reprove Gurvits’ theorem and discuss its importance. Gurvits’ originalproof of this fact was not very complicated, and our proof will be similar in this regard. This is of coursewhat makes capacity and real stability more generally so intriguing: answers to seemingly hard questionsfollow from a few basic computations on polynomials.9 heorem 3.1 (Gurvits) . For real stable p ∈ R λ + [ x , ..., x n ] we have: Cap (1 n − ) (cid:0) ∂ x k p | x k =0 (cid:1) Cap (1 n ) ( p ) ≥ (cid:18) λ k − λ k (cid:19) λ k − Proof.
We apply Theorem 4.11 above for T := ∂ x k | x k =0 , α := (1 n ), and β := (1 n − ). To do this we need tocompute the right-hand side of the expression in Theorem 4.11, making use of properties from Proposition2.14. α α ( λ − α ) λ − α λ λ Cap ( α,β ) (Symb λ ( T )) = ( λ − λ − λ λ Cap (1 n , n − ) (cid:16) ∂ x k (1 + xz ) λ (cid:12)(cid:12) x k =0 (cid:17) = ( λ − λ − λ λ Cap (1 n , n − ) λ k z k Y j = k (1 + x j z j ) λ j = ( λ − λ − λ λ λ k Y j = k Cap (1 , (cid:0) (1 + x j z j ) λ j (cid:1) Note that Cap (1 , ((1 + x j z j ) λ j ) = Cap ((1 + x j ) λ j ). Using the homogenization property and Lemma 2.16,we then have: Cap ((1 + x j ) λ j ) = Cap (1 ,λ j − (( x j + y j ) λ j ) = λ j (cid:18) λ j λ j − (cid:19) λ j − Therefore: α α ( λ − α ) λ − α λ λ Cap ( α,β ) (Symb λ ( T )) = ( λ − λ − λ λ λ k Y j = k Cap (1 , (cid:0) (1 + x j z j ) λ j (cid:1) = ( λ − λ − λ λ λ k Y j = k λ j (cid:18) λ j λ j − (cid:19) λ j − = (cid:18) λ k − λ k (cid:19) λ k − This proof will serve as a good baseline for other applications of our main theorems. Roughly speaking,most applications will make use of Lemma 2.16 and the properties of Proposition 2.14 in interesting ways.And often, the inequalities obtained will directly translate to various combinatorial statements.Specifically, what sorts of combinatorial statements can be derived from Gurvits’ theorem? The mostwell known are perhaps Schrijver’s theorem and the Van der Waerden bound on the permanent (see [Gur08]).What forms the link between capacity and combinatorial objects like doubly stochastic matrices and perfectmatchings is the following polynomial defined for a given matrix M : p M ( x ) := Y i X j m ij x j Note that this polynomial is real stable whenever the entries of M are nonnegative. The following is thenquite suggestive. Lemma 3.2 (Gurvits) . If M is a doubly stochastic matrix, then Cap (1 n ) ( p M ) = 1 .Proof. Follows from Fact 2.10. 10or most of the arguments, one considers p = p M and T such that T ( p ) computes the desired quantityrelated to M . The above theorems then give something like:desired quantity = Cap( T ( p M ))Cap( p M ) ≥ constant depending on Symb( T ) but not on M This gives us a bound on the desired quantity (e.g. perfect matchings or the permanent) for any M , so longas we can compute the capacity of Symb( T ).In addition to these types of inequalities, Gurvits also demonstrates how his theorem implies similarresults for “doubly stochastic” n -tuples of matrices (a conjecture due to Bapat [Bap89]). In fact, this notionof doubly stochastic aligns with a generalized notion used recently in [GGOW16],[BGO + The most important application of our results is a new proof of a bound on size- k matchings of a biregularbipartite graph, due to Csikv´ari [Csi14]. This result is a generalization of Schrijver’s bound, and it alsosettled and strengthened the Friedland macthing conjecture [FKM08]. We first state Csikv´ari’s results, in aform more amenable to the notation of this paper. Theorem 3.3 (Csikv´ari) . Let G be an ( a, b ) -biregular bipartite graph with ( m, n ) -bipartitioned vertices (sothat am = bn is the number of edges of G ). Then the number of size- k matchings of G is bounded as follows: µ k ( G ) ≥ (cid:18) nk (cid:19) ( ab ) k m m ( ma − k ) ma − k ( ma ) ma ( m − k ) m − k Notice that this immediately implies the following bound for regular bipartite graphs.
Corollary 3.4 (Csikv´ari) . Let G be a d -regular bipartite graph with n vertices. Then: µ k ( G ) ≥ (cid:18) nk (cid:19) d k (cid:18) nd − knd (cid:19) nd − k (cid:18) nn − k (cid:19) n − k To prove these results, we first need to generalize Gurvits’ capacity lemma for doubly stochastic matrices.Specifically we want to be able to handle ( a, b ) -stochastic matrices , which are matrices with row sums equalto a and columns sums equal to b . We care about such matrices, because the bipartite adjacency matrixof a ( a, b )-biregular graph is ( a, b )-stochastic. Note that if M is an ( a, b )-stochastic matrix which is of size m × n , then am = bn . Lemma 3.5. If M is an ( a, b ) -stochastic matrix, then Cap ( mn ,..., mn ) ( p M ) = a m .Proof. Follows from Fact 2.10.We also need a linear operator which computes the number of size- k matchings of an ( a, b )-biregularbipartite graph. In fact when M is the bipartite adjacency matrix of G , we have the following: a m − k µ k ( G ) = X S ∈ ( [ n ] k ) ∂ Sx p M (1) = Cap ∅ X S ∈ ( [ n ] k ) ∂ Sx p M (1) ! Note that each differential operator in the sum picks out a disjoint collection of k × k subpermutations ofthe matrix M . After applying each differential operator, we are left with terms which are products of m − k p M . Plugging in 1 then gives a m − k (since row sums are a ), and this is why thatfactor appears above.Next, we need to prove that we can apply Theorem 4.11 to the operator T := P S ∈ ( [ n ] k ) ∂ Sx (cid:12)(cid:12) x =1 . Wechoose λ = ( b, ..., b ) here because ( a, b )-regularity of G implies every variable will be of degree b in thepolynomial p M (where M is the bipartite adjacency matrix of G ). That is, some of the degree informationof G is encoded as the degree of the associated polynomial p M . This same thing was done in [Gur06] and[FG08], and this further demonstrates how capacity bounds can combine an interesting mix of analytic andcombinatorial information. Lemma 3.6.
The operator T := P S ∈ ( [ n ] k ) ∂ Sx (cid:12)(cid:12) x =1 has real stable symbol.Proof. Here the input polynomial space is R ( b,...,b )+ [ x , ..., x n ], since degree is determined by the column sums.Denoting λ := ( b, ..., b ), we compute Symb λ ( T ): T [(1 + xz ) λ ] = X S ∈ ( [ n ] k ) ∂ Sx (cid:12)(cid:12) x =1 (1 + xz ) λ = X S ∈ ( [ n ] k ) b k z S (1 + z ) λ − S = b k (1 + z ) λ − X S ∈ ( [ n ] k ) z S (1 + z ) − S Notice that P S ∈ ( [ n ] k ) z S (1 + z ) − S = (cid:0) nk (cid:1) Pol n ( x k (1 + x ) n − k ), which is real stable by Proposition 2.3.Applying Theorem 4.11 now shows us the way toward the rest of the proof. Denoting λ := ( b, ..., b ) and α := ( mn , ..., mn ), we now have: a m − k µ k ( G ) = X S ∈ ( [ n ] k ) ∂ Sx p M (1) ≥ α α ( λ − α ) λ − α λ λ Cap α ( p M ) Cap ( α, ∅ ) (Symb λ ( T ))= ( mn ) mn ( b − mn ) b − mn b b ! n a m Cap α (Symb λ ( T ))= ( ma ) m ( nb − m ) nb − m ( nb ) nb Cap α (Symb λ ( T ))So the last computation we need to make is that of Cap α (Symb λ ( T )). Fortunately since Symb λ ( T )is symmetric and α = ( mn , ..., mn ), we can apply the symmetric diagonalization property to simplify thiscomputation. Using our previous computation of Symb λ ( T ), this gives:Cap ( mn ,..., mn ) (Symb λ ( T )) = Cap m (cid:18) b k (cid:18) nk (cid:19) z k (1 + z ) nb − k (cid:19) = b k (cid:18) nk (cid:19) Cap m ( z k (1 + z ) nb − k )The remaining capacity computation then follows from homogenization and Lemma 2.16:Cap m ( z k (1 + z ) nb − k ) = inf z> z k (1 + z ) nb − k z m = Cap m − k ((1 + z ) nb − k ) = (cid:18) nb − km − k (cid:19) m − k (cid:18) nb − knb − m (cid:19) nb − m Putting all of these computations together and recalling ma = nb gives: µ k ( G ) ≥ a k − m ( ma ) m ( nb − m ) nb − m ( nb ) nb b k (cid:18) nk (cid:19) (cid:18) nb − km − k (cid:19) m − k (cid:18) nb − knb − m (cid:19) nb − m = (cid:18) nk (cid:19) a k − m b k ( ma ) m ( nb − k ) nb − k ( nb ) nb ( m − k ) m − k = (cid:18) nk (cid:19) ( ab ) k m m ( ma − k ) ma − k ( ma ) ma ( m − k ) m − k This is precisely the desired inequality. 12 .3 Differential Operators in General
We now give general capacity preservation bounds for stability preservers which are differential operators.This was done in [AG17] for differential operators which preserve real stability on input polynomials of alldegrees. Here, we restrict to those operators which only preserve real stability for polynomials of some fixedbounded degree. That said, consider the following bilinear operator:( p ⊞ λ q )( x ) := X ≤ µ ≤ λ ( ∂ µx p )( x )( ∂ λ − µx q )(0)It is straightforward to see that by fixing q , one can construct any constant coefficient differential operatoron R λ [ x , ..., x n ]. And it turns out that if q is real stable, then ( · ⊞ λ q )( x ) is a real stability preserver.It turns out that more is true, however. The operator ⊞ λ can actually be applied to polynomials in R ( λ,λ ) [ x , ..., x n , y , ..., y n ] by considering this polynomial space as a tensor product of polynomial spaces.More concretely, we specify how this operator acts on the monomial basis: ⊞ λ : x µ y ν x µ ⊞ λ x ν We can then compute the symbol of this operator:Symb[ ⊞ λ ] = (1 + xz ) λ ⊞ λ (1 + xw ) λ = ( z + w + zwx ) λ = ( zw ) λ ( x + z − + w − ) λ Note that Symb[ ⊞ λ ]( z, w, − x ) is real stable, and so ⊞ λ preserves real stability by Theorem 2.5.With this, we compute the capacity for λ = δ = (1 , , ..., ( α,β,γ ) ( z + t + ztx ) = inf x,z,t> z + t + ztxx α z β t γ = inf x,z,t> t − + z − + xx α z β − t γ − = inf x,z,t> t + z + xx α z − β t − γ Note that ( α, β, γ ) is in the Newton polytope of ( z + t + ztx ) iff α = β + γ −
1. By Lemma 2.16, we have:Cap ( α,β,γ ) ( z + t + ztx ) = 1 α α (1 − β ) − β (1 − γ ) − γ We now generalize this to general λ , supposing α = β + γ − λ :Cap ( α,β,γ ) (( z + t + ztx ) λ ) = n Y j =1 (cid:16) ( α j /λ j ) α j /λ j (1 − β j /λ j ) − β j /λ j (1 − γ j /λ j ) − γ j /λ j (cid:17) − λ j = n Y j =1 ( α j /λ j ) − α j (1 − β j /λ j ) β j − λ j (1 − γ j /λ j ) γ j − λ j = α − α ( λ − β ) β − λ ( λ − γ ) γ − λ λ α − β − γ +2 λ = λ λ α α ( λ − β ) λ − β ( λ − γ ) λ − γ Applying Theorem 4.11, we get:Cap α ( p ⊞ λ q ) ≥ β β γ γ ( λ − β ) λ − β ( λ − γ ) λ − γ λ λ λ λ · λ λ α α ( λ − β ) λ − β ( λ − γ ) λ − γ · Cap β ( p ) Cap γ ( q )= β β γ γ α α λ λ Cap β ( p ) Cap γ ( q )Again, this is all under the assumption that α = β + γ − λ . (We will be outside the Newton polytopeotherwise, and so the result in that case will be trivial.) We state the result of this discussion as follows.Note that is can be seen as a sort of multiplicative reverse triangle inequality for capacity of differentialoperators. 13 orollary 3.7. Let p, r be two real stable polynomials of degree λ with positive coefficients. We have: ( α α Cap α ( p ⊞ λ q )) ≥ λ λ ( β β Cap β ( p ))( γ γ Cap γ ( q ))With this, we have given tight capacity bounds for all differential operators on polynomials of at most somefixed bounded degree. Note that root bounds of this form are given in [MSS15a] by Marcus, Spielman, andSrivastava, and these bounds are related to those obtained in their proof of Kadison-Singer in [MSS15b]. It isan open and interesting question whether or not capacity can be utilized to bound the roots of polynomials. We now discuss our main results and the inequalities we use to obtain them. These inequalities are boundson certain inner products applied to polynomials. The most basic of these is the main result from [AG17],which applies to multiaffine polynomials. We extend their methods to obtain bounds on polynomials of alldegrees. Finally, a limiting argument implies bounds for the LP + class. This last bound can also be foundin [AG17], but the proof we give here is simpler and makes clearer the connection between these inequalitiesand the Borcea-Br¨and´en characterization. For polynomials of some fixed bounded degree, we consider the following inner product.
Definition 4.1.
For fixed λ ∈ Z n + and p, q ∈ R λ [ x , ..., x n ], define: h p, q i λ := X ≤ µ ≤ λ (cid:18) λµ (cid:19) − p µ q µ As mentioned above, Anari and Gharan prove a bound on the above inner product for multiaffine polyno-mials in [AG17], and we state their result here without proof. We note though that the proof is essentially aconsequence of the strong Rayleigh inequalities for real stable polynomials, which we now state. These fun-damental inequalities (due to Br¨and´en) should be seen as log-concavity conditions, and this intuition extendsto all the inner product bounds we state here. And this intuition is not without evidence: the connectionof capacity to the Alexandrov-Fenchel inequalities (see [Gur06]), as well as to matroids and log-concavepolynomials (see [Gur09] and more recently [AGV18]), has been previously noted.
Proposition 4.2 (Strong Rayleigh inequalities [Br¨a07]) . For any real stable p ∈ R (1 n ) and any i, j ∈ [ n ] ,we have the following inequality pointwise on all of R n : ( ∂ x i p ) · ( ∂ x j p ) ≥ p · ( ∂ x i ∂ x j p )We now state the Anari-Gharan bound for multiaffine polynomials. They also prove a weaker bound onpolynomials of any degree, but we will discuss this later. Theorem 4.3 (Anari-Gharan) . Let p, q ∈ R (1 n )+ [ x , ..., x n ] be real stable. Then for any α ∈ R n + we have: h p, q i (1 n ) ≥ α α (1 − α ) − α Cap α ( p ) Cap α ( q )In this paper, we generalize this to polynomials of degree λ as follows. Note that this result is strictlystronger than the bound obtained in [AG17] for the non-multiaffine case. Theorem 4.4.
Let p, q ∈ R λ + [ x , ..., x n ] be real stable. Then for any α ∈ R n + we have: h p, q i λ ≥ α α ( λ − α ) λ − α λ λ Cap α ( p ) Cap α ( q )14he proof of this is essentially due to the fact that both h· , ·i λ and capacity interact nicely with polariza-tion. We have already explicated the connection between capacity and polarization (see Proposition 2.15),and we now demonstrate how these inner products fit in. Lemma 4.5.
Given p, q ∈ R λ [ x , ..., x n ] , we have: h p, q i λ = D Pol λ ( p ) , Pol λ ( q ) E (1 λ ) Proof.
We compute this equality on a basis in the univariate case. The result then follows since Pol λ is acomposition of polarizations on each variable of p . For 0 ≤ k ≤ m we have: (cid:10) Pol m ( x k ) , Pol m ( x k ) (cid:11) (1 m ) = (cid:18) mk (cid:19) − X S ∈ ( [ m ] k ) h x S , x S i (1 m ) = (cid:18) mk (cid:19) − = h x k , x k i m The proof of Theorem 4.4 then essentially follows from this algebraic identity.
Proof of Theorem 4.4.
Suppose that p, q ∈ R λ + [ x , ..., x n ] are real stable polynomials. Then Pol λ ( p ) andPol λ ( q ) are real stable multiaffine polynomials by Proposition 2.3. We now use the multiaffine bound toprove the result for any α ∈ R n + . For simplicity, let β := Pol λ ( α ), where Pol λ ( α ) is originally defined in § h p, q i λ = D Pol λ ( p ) , Pol λ ( q ) E (1 λ ) ≥ β β (1 − β ) − β Cap β (Pol λ ( p )) Cap β (Pol λ ( q ))By Proposition 2.15, we have that Cap β (Pol λ ( p )) = Cap α ( p ). So to complete the proof, we compute: β β (1 − β ) − β = n Y k =1 λ k Y j =1 (cid:18) α k λ k (cid:19) α k /λ k (cid:18) − α k λ k (cid:19) − α k /λ k = n Y k =1 (cid:18) α k λ k (cid:19) α k (cid:18) λ k − α k λ k (cid:19) λ k − α k This is precisely α α ( λ − α ) λ − α λ λ , which is what was claimed. For general polynomials and power series in the LP + class, we consider the following inner product. Definition 4.6.
For p, q ∈ R [ x , ..., x n ] or power series in x , ..., x n , define: h p, q i ∞ := X ≤ µ µ ! p µ q µ Note that this may not be well-defined for some power series.Consider the following power series in x , ..., x n , where c µ ≥ f ( x , ..., x n ) = X ≤ µ µ ! c µ x µ Next consider the following weighted truncations of f : f λ ( x ) := X ≤ µ ≤ λ (cid:18) λµ (cid:19) c µ x µ f ∈ LP + [ x , ..., x n ], then f λ is real stable for all λ and f λ ( x/λ ) → f ( x ) uniformly on compact sets in C n (see Theorem 5.1 in [BB09a]). The idea then is to limit capacity bounds for polynomials of some boundeddegree to capacity bounds for general polynomials and functions in the LP + class.To do this, we need some kind of continuity result for capacity. Note that Fact 2.13 implies Cap α ( p ) is not continuous in α at the boundary of the Newton polytope of p . However, it turns out Cap α ( p ) is continuousin p , for the topology of uniform convergence on compact sets. This is discussed in § Corollary 5.8.
Let p n be polynomials with nonnegative coefficients and p analytic such that p n → p uni-formly on compact sets. For α ∈ Newt( p ) , we have: lim n →∞ Cap α p n = Cap α p We now demonstrate the link between the bounded and unbounded degree inner products, and we willuse this to obtain bounds on the latter via limiting.
Lemma 4.7.
Let f and f λ be defined as above. For any p ∈ R + [ x , ..., x n ] we have: lim λ →∞ h f λ , p i λ = h f, p i ∞ Proof.
Letting c µ denote the weighted coefficients of f and f λ as above, we compute:lim λ →∞ h f λ , p i λ = lim λ →∞ X ≤ µ ≤ λ c µ p µ = X ≤ µ c µ p µ = h f, p i ∞ Notice that the limit here is guaranteed to exist, since p has finite support.With this, we can bootstrap our capacity bound on h· , ·i λ to get a bound on h· , ·i ∞ . Notice here that weachieve the same bound as Anari and Gharan in [AG17], albeit with a simpler proof. Theorem 4.8 (Anari-Gharan) . Fix f ∈ LP + [ x , ..., x n ] and any real stable p ∈ R + [ x , ..., x n ] . Then forany α ∈ R n + we have: h f, p i ∞ ≥ α α e − α Cap α ( f ) Cap α ( p ) Proof.
As above, we write: f ( x ) = X ≤ µ µ ! c µ x µ f λ ( x ) = X ≤ µ ≤ λ (cid:18) λµ (cid:19) c µ x µ By the previous lemma, we have: h f, p i ∞ = lim λ →∞ h f λ , p i λ ≥ lim λ →∞ (cid:20) α α ( λ − α ) λ − α λ λ Cap α ( f λ ) Cap α ( p ) (cid:21) = α α Cap α ( p ) · lim λ →∞ (cid:20) ( λ − α ) λ − α λ λ · inf x> f λ ( x/λ )( x/λ ) α (cid:21) = α α Cap α ( p ) · lim λ →∞ (cid:20) ( λ − α ) λ − α λ λ − α · Cap α ( f λ ( x/λ )) (cid:21) Notice that lim λ →∞ Cap α ( f λ ( x/λ )) = Cap α ( f ) by Corollary 5.8. So we just need to compute the limit ofthe scaling factor: lim λ →∞ (cid:18) λ − αλ (cid:19) λ − α = lim λ →∞ n Y k =1 (cid:18) − α k λ k (cid:19) λ k − α k = n Y k =1 e − α k = e − α This completes the proof.
Corollary 4.9.
Fix f, g ∈ LP + [ x , ..., x n ] . For any α ∈ R n + we have: h f, g i ∞ ≥ α α e − α Cap α ( f ) Cap α ( g ) Proof.
Apply the previous theorem and Corollary 5.8 to a sequence of real stable polynomials g k → g .16 .3 From Inner Products to Linear Operators The main purpose of this section, aside from proving the main technical result of the paper, is to demonstratethe power of a certain interpretation of the symbol of a linear operator. We will show that a simpleobservation regarding the symbol (which is explicated in more detail in [Lea17]) will immediately enable usto transfer inner product bounds to bounds on linear operators. We now state this observation, which couldbe considered as a more algebraic definition of the symbol.
Lemma 4.10.
Let h· , ·i be either h· , ·i λ or h· , ·i ∞ , and let Symb be either
Symb λ or Symb ∞ , respectively.Let T be a linear operator on polynomials of appropriate degree, and let p, q ∈ R + [ x , ..., x n ] be polynomialsof appropriate degree. Then we have the following, where the inner product acts on the z variables: T [ p ]( x ) = h Symb[ T ]( z, x ) , p ( z ) i Proof.
Straightforward, as the scalars present in the expressions of h· , ·i and Symb were chosen such thatthey cancel out in the above expression. One could compute this on the monomial basis, for example.As we will see very shortly, this will make for quick proofs of the main results given the inner productbounds we have already achieved. Before doing this though, let us discuss some of the linear operator boundsthat Anari and Gharan achieved in [AG17]. Note the following differential operator form of h· , ·i ∞ : h p, q i ∞ = q ( ∂ x ) q ( x ) | x =0 Anari and Gharan then use use their inner product bound to essentially give capacity preservation results forcertain differential operators. Similarly, for multiaffine polynomials h p, q i (1 n ) = q ( ∂ x ) q ( x ) | x =0 , which givesa better bound in the multiaffine case. We now vastly generalize this idea, with a rather short proof. Theorem 4.11.
Let T : R λ + [ x , ..., x n ] → R γ + [ x , ..., x m ] be a linear operator such that Symb λ ( T ) is realstable in z for every x ∈ R m + . Then for any real stable p ∈ R λ + [ x , ..., x n ] , any α ∈ R n + , and any β ∈ R m + wehave: Cap β ( T ( p ))Cap α ( p ) ≥ α α ( λ − α ) λ − α λ λ Cap ( α,β ) (Symb λ ( T )) Further, this bound is tight for fixed T , α , and β .Proof. In the proof, let h· , ·i := h· , ·i λ and Symb := Symb λ . By the previous lemma, we have the followingfor any fixed x ∈ R n + . (Here, the inner product acts on the z variables.) T ( p )( x ) = h Symb( T )( z, x ) , p ( z ) i Theorem 4.4 then implies: T ( p )( x ) = h Symb( T )( z, x ) , p ( z ) i ≥ α α ( λ − α ) λ − α λ λ Cap α ( p ) · Cap α (Symb( T )( · , x ))Dividing by x β on both sides and taking inf gives:inf x > T ( p )( x ) x β ≥ α α ( λ − α ) λ − α λ λ Cap α ( p ) · inf x > inf z> Symb( T )( z, x ) z α x β This is the desired result. Tightness then follows from considering input polynomials of the form p ( x ) = Q k (1 + x k y k ) for fixed y ∈ R n + , and then taking inf over y .As stated in the introduction, this is our main technical result, and we have already discussed some ofits applications in §
3. We give a similar result for linear operators on polynomials of any degree.17 heorem 4.12.
Let T : R + [ x , ..., x n ] → R + [ x , ..., x m ] be a linear operator such that Symb ∞ ( T ) is in LP + [ z , ..., z n ] for every x ∈ R m + . Then for any p ∈ R + [ x , ..., x n ] , any α ∈ R n + , and any β ∈ R m + we have: Cap β ( T ( p ))Cap α ( p ) ≥ e − α α α Cap ( α,β ) (Symb ∞ ( T )) Further, this bound is tight for fixed T , α , and β .Proof. The proof given above for Theorem 4.11 can be essentially copied verbatim.We now combine these results with the Borcea-Br¨and´en characterization results (Theorems 2.5 and 2.8)to give concrete corollaries which directly relate to stability preservers.
Corollary 4.13.
Suppose T : R λ + [ x , ..., x n ] → R γ + [ x , ..., x m ] is a linear operator of rank greater than 2,such that T preserves real stability. Then Theorem 4.11 applies to T .Proof. Since the image of T is of dimension greater than 2, Theorem 2.5 implies one of two possibilities:1. Symb λ [ T ] is real stable.2. Symb λ [ T ]( z , ..., z n , − x , ..., − x n ) is real stable.In either case, we have that Symb λ [ T ] is real stable in z for every fixed x ∈ R m + (see Proposition 2.1).Therefore Theorem 4.11 applies. Corollary 4.14.
Suppose T : R + [ x , ..., x m ] → R + [ x , ..., x m ] is a linear operator of rank greater than 2,such that T preserves real stability. Then Theorem 4.12 applies to T .Proof. The same proof works, using Theorem 2.8 instead.
In this section, we discuss the continuity of capacity as a function of the the input polynomial p . The mainresult of this section allows us to limit inner product bounds from h· , ·i λ to h· , ·i ∞ , which is exactly how weproved Theorem 4.8.Given a (positive) discrete measure µ on R n , we define its generating function as: p µ ( x ) := X κ ∈ supp( µ ) µ ( κ ) x κ (Note that we have only restricted supp( µ ) to be in R n , and so p µ may not be a polynomial.) We furtherdefine the log-generating function of µ as: P µ ( x ) := log( p µ (exp( x ))) = log X κ ∈ supp( µ ) µ ( κ ) exp( x · κ )More generally for such a function p ( x ), we will write: p ( x ) := X κ p κ x κ P ( x ) := log( p (exp( x ))) = log X κ p κ exp( x · κ )We care about discrete measures (with not necessarily finite support) whose generating functions are conver-gent and continuous on R n + . This is equivalent to the log-generating function being continuous on R n . Notethat an important example of such a measure is one which has finite support entirely in Z n + . The generatingfunctions of such measures are polynomials.From now on we will write supp( p ) = supp( P ) to denote the support of µ (as above) and Newt( p ) =Newt( P ) to denote the polytope generated by its support. We first give a few basic results.18 act 2.13. For p a continuous generating function, the following are equivalent.1. α ∈ Newt( p ) Cap α p ( x ) > P ( x ) − α · x is bounded below. Lemma 5.1.
Any continuous log-generating function Q ( x ) is convex in R n .Proof. H¨older’s inequality.Note that proving statements for p is essentially the same as proving for P , as suggested in the followinglemma. Lemma 5.2.
Let p, p n be continuous generating functions. Then p n → p uniformly on compact sets of R n + iff P n → P uniformly on compact sets of R n .Proof. Equivalence of p n → p and exp( P n ) → exp( P ) follows form the fact that exp : R n + → R n is ahomeomorphism (and so gives a bijection of compact sets). The fact that exp and log are (uniformly)continuous on every compact set in their domains then completes the proof.We now get the first half of the desired equality, which is the easier half. Lemma 5.3.
With p, p n continuous generating functions and p n → p uniformly on compact sets, we have: lim n →∞ inf p n ≤ inf p Proof.
Let ( x m ) ⊂ R n + be a sequence such that p ( x m ) → inf p . For each m we have that p n ( x m ) is eventuallynear to p ( x m ). So for any fixed ǫ >
0, we have the following for m = m ( ǫ ) and n ≥ N ( ǫ, m ):inf p n ≤ p n ( x m ) ≤ p ( x m ) + ǫ ≤ inf p + 2 ǫ The result follows by sending ǫ → α is on the boundary of Newt( p ). Lemma 5.4.
Suppose is in the interior of Newt( p ) . Then inf P is attained precisely on some compactconvex subset K of R n .Proof. By a previous lemma, inf P is finite. Suppose x n is an unbounded sequence (with monotonicallyincreasing norm) such that P ( x n ) limits to inf P . By compactness of the n -dimensional sphere, we canassume by restricting to a subsequence that x n || x n || limits to some u . Pick ǫ > ǫu ∈ Newt( p ), and consider P ( x ) − ǫu · x . We then have:lim n →∞ P ( x n ) − ǫu · x n = lim n →∞ P ( x n ) − ǫ || x n || (cid:18) u · x n || x n || (cid:19) = −∞ However, since ǫu ∈ Newt( p ) we have that P ( x ) − ǫu · x is bounded below, a contradiction. So, every sequencelimiting to inf P is bounded, and therefore inf P is attained on a bounded set. By convexity of P , this setis convex.The next few results then finish the proof of continuity of Cap α ( · ) under certain support conditions. Proposition 5.5.
Let p and p n be continuous generating functions such that p n → p , with in the interiorof Newt( p ) . Then: lim n →∞ inf p n = inf p roof. Given the above lemma, we only have the ≥ direction left to prove. Since 0 is in the interior ofNewt( p ), there is some compact convex K ⊂ R n such that P ( x ) = inf P iff x ∈ K . Further, this implies thatfor any compact set K ′ whose interior contains K , there exists c > P ( x ) > inf P + c on theboundary of K ′ . For any fixed positive ǫ < c and large enough n , we then have: | P n − P | < ǫ < c K ′ = ⇒ | P n − inf P | < ǫ < c KP n > inf P + ( c − ǫ ) > inf P + c K ′ Convexity of P n then implies P n ( x ) > inf P + c outside of K ′ . Therefore for any ǫ and large enough n :inf P n = inf x ∈ K ′ P n ≥ inf P − ǫ Letting ǫ → p ). This ends upneeding a bit more restriction. Lemma 5.6.
Suppose is on the boundary on Newt( P ) . Then there exists A ∈ SO n ( R ) such that: Newt( A · P ) ⊂ { κ : κ n ≥ } inf ( A · P ) | x n = −∞ = inf P Proof.
Since 0 is on the boundary of the convex set Newt( P ), a separating hyperplane gives a unit vector c such that ( c | µ ) ≥ µ ∈ Newt( P ). Let A ∈ SO n ( R ) be such that Ac = e n . We first have:inf A · P = inf P ( A − x ) = inf P Since Newt( A · P ) = A · Newt( P ) and ( e n | Aµ ) = ( c | µ ) ≥ µ ∈ Newt( P ), we have that Newt( A · P ) ⊂{ κ : κ n ≥ } . Therefore: inf ( A · P ) | x n = −∞ = inf A · P = inf P Note that ( A · P ) | x n = −∞ denotes the continuous log-generating function given by the terms κ of the supportof A · P for which κ n = 0. This is justified, as Newt( A · P ) ⊂ { κ : κ n ≥ } implies that A · P decreases as x n decreases (and we care about inf). Theorem 5.7.
Let p and p m be continuous generating functions such that p m → p , with ∈ Newt( p ) .Suppose further that eventually Newt( p m ) ⊆ Newt( p ) . Then: lim m →∞ inf p m = inf p Proof.
Given the above proposition, we only need to prove this in the case where 0 is on the boundaryof Newt( p ). In that case, the previous lemma gives an A ∈ SO n ( R ) such that Newt( A · P ) ⊂ { κ : κ n ≥ } and inf ( A · P ) | x n = −∞ = inf P . Since P m → P implies A · P m → A · P , we now relax to provinglim m →∞ inf A · P m = inf A · P . By assumption, eventually Newt( P m ) ⊆ Newt( P ) which implies Newt( A · P m ) ⊆ Newt( A · P ) ⊂ { κ : κ n ≥ } . So, eventually Newt( ( A · P m ) | x n = −∞ ) ⊆ Newt( ( A · P ) | x n = −∞ ) andinf A · P m = inf ( A · P m ) | x n = −∞ . By induction on the number of variables, we then have:lim m →∞ inf A · P m = lim m →∞ inf ( A · P m ) | x n = −∞ = inf ( A · P ) | x n = −∞ = inf A · P For the base case, p m and p are scalars and the result is trivial. Corollary 5.8.
Let p n be polynomials with nonnegative coefficients and p analytic such that p n → p , with α ∈ Newt( p ) . Then: lim n →∞ Cap α p n = Cap α p roof. As in the previous proposition, we only have the ≥ direction to prove. Let q n be defined as the sum ofthe terms of p n which appear in the support of p . Since the p n are polynomials with nonnegative coefficients,we have that q n → p . By the previous theorem, we then have:lim n →∞ Cap α p n ≥ lim n →∞ Cap α q n = Cap α p Note that the fact that q n → p holds after restricting to the support of p relies on the fact that p n and q n are polynomials with positive coefficients. This is the main barrier to generalizing this corollary to allcontinuous generating functions. We have given here tight bounds on capacity preserving operators related to real stable polynomials. Theseresults are essentially corollaries of inner product bounds, extended from bounds of Anari and Gharan, alleventually based on the strong Rayleigh inequalities. That said, there are a number of pieces of this thatmay be able to be altered or generalized, and this raises new questions.The first is that of the inner product: are there other inner products for which we can obtain bounds?The main conjecture in this direction is that of Gurvits in [Gur09].
Conjecture 6.1 (Gurvits) . Let p, q ∈ R + [ x , ..., x n ] be homogeneous real stable polynomials of total degree d . Then: X || µ | = d (cid:18) dµ (cid:19) − p µ q µ ≥ α α d d Cap α ( p ) Cap α ( q )The main difference here is that we use multinomial coefficients rather than products of binomial coeffi-cients. Note that the symbol operator associated to this inner product is given by T [( z · x ) d ] (dot product of z and x ). It is not immediately clear how this inner product relates to real stable polynomials, as the linkto stability preservers is less clear than in the Borcea-Br¨and´en case.The next is the class of polynomials: are there more general classes of polynomials for which weakercapacity bounds can be achieved? One such bound is achieved for strongly log-concave polynomials (originallystudied by Gurvits in [Gur09]) in [AGV18], and this class contains basis generating polynomials of matroids.(Note that the authors call these polynomials completely log-concave , and they are also called Lorentzian in[BH19].) This bound relies on a weakened version of the strong Rayleigh inequalities, where a factor of 2 isintroduced. It is unclear what applications such a bound has beyond those of [AGV18].The last is a question about the further applicability of the main results of this paper. In particular, allof the operators studied here are differential operators. Are there applications of non-differential operators?Also, are there ways to get a handle on the location of the roots of a polynomial via capacity? This secondquestion is of particular interest, as it may lead to a more unified and a direct approach to various rootbounding results. For example, the root bounds of [MSS15a] are at the heart of the proof of Kadison-Singerin [MSS15b]. Can capacity be used to achieve those bounds?
References [AG17] Nima Anari and Shayan Oveis Gharan,
A generalization of permanent inequalities and applica-tions in counting and optimization , Proceedings of the 49th Annual ACM SIGACT Symposiumon Theory of Computing, ACM, 2017, pp. 384–396.[AGV18] Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzant,
Log-concave polynomials, entropy,and a deterministic approximation algorithm for counting bases of matroids , arXiv preprintarXiv:1807.00929 (2018). 21AHK18] Karim Adiprasito, June Huh, and Eric Katz,
Hodge theory for combinatorial geometries , Annalsof Mathematics (2018), no. 2, 381–452.[Bap89] RB Bapat,
Mixed discriminants of positive semidefinite matrices , Linear Algebra and its Appli-cations (1989), 107–124.[BB09a] Julius Borcea and Petter Br¨and´en,
The lee-yang and p´olya-schur programs. i. linear operatorspreserving stability , Inventiones mathematicae (2009), no. 3, 541–569.[BB09b] ,
The lee-yang and p´olya-schur programs. ii. theory of stable polynomials and applications ,Communications on Pure and Applied Mathematics (2009), no. 12, 1595—-1631.[BGO +
17] Peter B¨urgisser, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson,
Alternat-ing minimization, scaling algorithms, and the null-cone problem from invariant theory , arXivpreprint arXiv:1711.08039 (2017).[BH19] Petter Br¨and´en and June Huh,
Lorentzian polynomials , arXiv preprint arXiv:1902.03719 (2019).[Br¨a07] Petter Br¨and´en,
Polynomials with the half-plane property and matroid theory , Advances in Math-ematics (2007), no. 1, 302–320.[CC89] Thomas Craven and George Csordas,
Jensen polynomials and the tur´an and laguerre inequalities ,Pacific Journal of Mathematics (1989), no. 2, 241–260.[COSW04] Young-Bin Choe, James G Oxley, Alan D Sokal, and David G Wagner,
Homogeneous multi-variate polynomials with the half-plane property , Advances in Applied Mathematics (2004),no. 1-2, 88–187.[Csi14] P´eter Csikv´ari, Lower matching conjecture, and a new proof of schrijver’s and gurvits’s theorems ,arXiv preprint arXiv:1406.0766 (2014).[Ero81] GP Erorychev,
Proof of the van der waerden conjecture for permanents , Siberian MathematicalJournal (1981), no. 6, 854–859.[Fal81] Dmitry I Falikman, Proof of the van der waerden conjecture regarding the permanent of a doublystochastic matrix , Mathematical notes of the Academy of Sciences of the USSR (1981), no. 6,475–479.[FG08] Shmuel Friedland and Leonid Gurvits, Lower bounds for partial matchings in regular bipartitegraphs and applications to the monomer–dimer entropy , Combinatorics, Probability and Com-puting (2008), no. 3, 347–361.[FKM08] Shmuel Friedland, E Krop, and Klas Markstr¨om, On the number of matchings in regular graphs ,the electronic journal of combinatorics (2008), no. 1, 110.[GGOW15] Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson, Operator scaling: theory andapplications , arXiv preprint arXiv:1511.03730 (2015).[GGOW16] ,
A deterministic polynomial time algorithm for non-commutative rational identity test-ing , Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, IEEE,2016, pp. 109–117.[Gur06] Leonid Gurvits,
Hyperbolic polynomials approach to van der waerden/schrijver-valiant like con-jectures: sharper bounds, simpler proofs and algorithmic applications , Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, ACM, 2006, pp. 417–426.22Gur08] ,
Van der waerden/schrijver-valiant like conjectures and stable (aka hyperbolic) homoge-neous polynomials: one theorem for all , the electronic journal of combinatorics (2008), no. 1,66.[Gur09] , On multivariate newton-like inequalities , Advances in Combinatorial Mathematics,Springer, 2009, pp. 61–78.[Gur11] ,
Unleashing the power of schrijver’s permanental inequality with the help of the betheapproximation , arXiv preprint arXiv:1106.2844 (2011).[HL72] Ole J. Heilmann and Elliott H. Lieb,
Theory of monomer-dimer systems , Communications inMathematical Physics (1972), 190–232.[HSW18] June Huh, Benjamin Schr¨oter, and Botong Wang,
Correlation bounds for fields and matroids ,arXiv preprint arXiv:1806.02675 (2018).[Lea17] Jonathan Leake,
A representation theoretic explanation of the borcea-br¨and´en characterization,and an extension to interval-rooted polynomials , arXiv preprint arXiv:1706.06168 (2017).[MSS15a] A Marcus, D Spielman, and N Srivastava,
Finite free convolutions of polynomials , arXiv preprint(2015), arXiv:1504.00350.[MSS15b] Adam W Marcus, Daniel A Spielman, and Nikhil Srivastava,
Interlacing families ii: Mixedcharacteristic polynomials and the kadison—singer problem , Annals of Mathematics (2015), 327–350.[Ren06] James Renegar,
Hyperbolic programs, and their derivative relaxations , Foundations of Compu-tational Mathematics (2006), no. 1, 59–79.[Sch98] Alexander Schrijver, Counting 1-factors in regular bipartite graphs , J. Comb. Theory, Ser. B (1998), no. 1, 122–135.[SV17] Damian Straszak and Nisheeth K Vishnoi, Real stable polynomials and matroids: Optimiza-tion and counting , Proceedings of the 49th Annual ACM SIGACT Symposium on Theory ofComputing, ACM, 2017, pp. 370–383.[Wal22] JL Walsh,