Completely log-concave polynomials and matroids ⇒ Cynthia Vinzant (North Carolina State University) joint work with Nima Anari, KuiKui Liu, Shayan Oveis Gharan (Stanford) (U. Washington) (U. Washington) Cynthia Vinzant Completely log-concave polynomials and matroids
Matroids A matroid on ground set [ n ] = { 1 , . . . , n } is a nonempty collection I of independent subsets of [ n ] satisfying: ◮ If S ⊆ T and T ∈ I , then S ∈ I . ◮ If S , T ∈ I and | T | > | S | , then ∃ i ∈ T \ S with S ∪ { i } ∈ I . Examples: ◮ linear independence of vectors v 1 , . . . , v n ∈ R d ◮ cyclic independence of n edges in a graph I ∈I y n −| I | � Independence poly. g M ( y , z 1 , . . . , z n ) = � i ∈ I z i Cynthia Vinzant Completely log-concave polynomials and matroids
Results and other work Mason’s conjecture: Let I k = # indep. sets of matroid M of size k . (i) I 2 k ≥ I k − 1 · I k +1 (log-concavity) � k +1 (ii) I 2 � k ≥ · I k − 1 · I k +1 k � 2 � I k ≥ I k − 1 � · I k +1 (iii) (ultra log-concavity) � n � n � n � � k k − 1 k +1 Cynthia Vinzant Completely log-concave polynomials and matroids
Results and other work Mason’s conjecture: Let I k = # indep. sets of matroid M of size k . (i) I 2 k ≥ I k − 1 · I k +1 (log-concavity) � k +1 (ii) I 2 � k ≥ · I k − 1 · I k +1 k � 2 � I k ≥ I k − 1 � · I k +1 (iii) (ultra log-concavity) � n � n � n � � k k − 1 k +1 Adiprasito, Huh, Katz use combinatorial Hodge theory to prove (i) Huh, Schr¨ oter, Wang use ↑ to prove (ii) Anari, Liu, Oveis Gharan, V. use complete log-concavity to prove (iii) Br¨ and´ en, Huh independently use Lorentz polynomials to prove (iii) Cynthia Vinzant Completely log-concave polynomials and matroids
Complete log-concavity f ∈ R [ z 1 , . . . , z n ] is log-concave on R n > 0 if f ≡ 0 or f ( x ) ≥ 0 for all x ∈ R n log( f ) is concave on R n and > 0 . ≥ 0 Cynthia Vinzant Completely log-concave polynomials and matroids
Complete log-concavity f ∈ R [ z 1 , . . . , z n ] is log-concave on R n > 0 if f ≡ 0 or f ( x ) ≥ 0 for all x ∈ R n log( f ) is concave on R n and > 0 . ≥ 0 For v = ( v 1 , . . . , v n ) ∈ R n , let D v = � n i =1 v i ∂ f ∂ z i . f ∈ R [ z 1 , . . . , z n ] is completely log-concave (CLC) on R n > 0 if for all k ∈ N , v 1 , . . . , v k ∈ R n ≥ 0 , D v 1 · · · D v k f is log-concave on R n ≥ 0 . Cynthia Vinzant Completely log-concave polynomials and matroids
Complete log-concavity f ∈ R [ z 1 , . . . , z n ] is log-concave on R n > 0 if f ≡ 0 or f ( x ) ≥ 0 for all x ∈ R n log( f ) is concave on R n and > 0 . ≥ 0 For v = ( v 1 , . . . , v n ) ∈ R n , let D v = � n i =1 v i ∂ f ∂ z i . f ∈ R [ z 1 , . . . , z n ] is completely log-concave (CLC) on R n > 0 if for all k ∈ N , v 1 , . . . , v k ∈ R n ≥ 0 , D v 1 · · · D v k f is log-concave on R n ≥ 0 . log( f ) ′′ = � d Example: f = � d − 1 i =1 ( z + r i ) ⇒ ( z + r i ) 2 ≤ 0 i =1 Cynthia Vinzant Completely log-concave polynomials and matroids
Example: stable polynomials f ∈ R [ z 1 , . . . , z n ] d is stable if f ( tv + w ) ∈ R [ t ] is real rooted for all v ∈ R n ≥ 0 , w ∈ R n . ⇒ f is completely log-concave Cynthia Vinzant Completely log-concave polynomials and matroids
Example: stable polynomials f ∈ R [ z 1 , . . . , z n ] d is stable if f ( tv + w ) ∈ R [ t ] is real rooted for all v ∈ R n ≥ 0 , w ∈ R n . ⇒ f is completely log-concave Example: det( � n i =1 z i v i v T d ) det( v i : i ∈ I ) 2 � i ) = � i ∈ I z i I ∈ ( [ n ] Cynthia Vinzant Completely log-concave polynomials and matroids
Example: stable polynomials f ∈ R [ z 1 , . . . , z n ] d is stable if f ( tv + w ) ∈ R [ t ] is real rooted for all v ∈ R n ≥ 0 , w ∈ R n . ⇒ f is completely log-concave Example: det( � n i =1 z i v i v T d ) det( v i : i ∈ I ) 2 � i ) = � i ∈ I z i I ∈ ( [ n ] Choe, Oxley, Sokal, Wagner: If f = � d ) c I � i ∈ I z i is stable, I ∈ ( [ n ] then supp ( f ) = { I : c I � = 0 } are the bases of a matroid on [ n ]. Br¨ and´ en: Fano matroid � = support of a stable polynomial f Cynthia Vinzant Completely log-concave polynomials and matroids
Equivalent conditions and univariate characterization Gurvits: f is strongly log-concave (SLC) if ∂ z 1 ) α 1 · · · ( ∂ ∂ α f = ( ∂ ∂ z n ) α n f is log-concave on R n ≥ 0 . Cynthia Vinzant Completely log-concave polynomials and matroids
Equivalent conditions and univariate characterization Gurvits: f is strongly log-concave (SLC) if ∂ z 1 ) α 1 · · · ( ∂ ∂ α f = ( ∂ ∂ z n ) α n f is log-concave on R n ≥ 0 . Theorem (ALOV): For f ∈ R [ z 1 , . . . , z n ] d , � ∂ α f is indecomposable for all | α | ≤ d − 2 f CLC ⇔ f SLC ⇔ and ∂ α f is CLC for all | α | = d − 2 Cynthia Vinzant Completely log-concave polynomials and matroids
Equivalent conditions and univariate characterization Gurvits: f is strongly log-concave (SLC) if ∂ z 1 ) α 1 · · · ( ∂ ∂ α f = ( ∂ ∂ z n ) α n f is log-concave on R n ≥ 0 . Theorem (ALOV): For f ∈ R [ z 1 , . . . , z n ] d , � ∂ α f is indecomposable for all | α | ≤ d − 2 f CLC ⇔ f SLC ⇔ and ∂ α f is CLC for all | α | = d − 2 (d=2) f = z T Qz is CLC ⇔ Q ij ≥ 0 and Q has 1 pos. eig. value. Cynthia Vinzant Completely log-concave polynomials and matroids
Equivalent conditions and univariate characterization Gurvits: f is strongly log-concave (SLC) if ∂ z 1 ) α 1 · · · ( ∂ ∂ α f = ( ∂ ∂ z n ) α n f is log-concave on R n ≥ 0 . Theorem (ALOV): For f ∈ R [ z 1 , . . . , z n ] d , � ∂ α f is indecomposable for all | α | ≤ d − 2 f CLC ⇔ f SLC ⇔ and ∂ α f is CLC for all | α | = d − 2 (d=2) f = z T Qz is CLC ⇔ Q ij ≥ 0 and Q has 1 pos. eig. value. Cor. (Gurvits/ALOV) � 2 n � a k ≥ a k − 1 � · a k +1 a k y n − k z k is CLC ⇔ � � n � n � n � � k − 1 k +1 k k =0 Cynthia Vinzant Completely log-concave polynomials and matroids
Complete log-concavity for matroids I ∈I y n −| I | � Theorem. g M ( y , z 1 , . . . , z n ) = � i ∈I z i is CLC. (just check rank-two matroids M ) k =0 I k y n − k z k is CLC. Cor: g M ( y , z , . . . , z ) = � n Cor: {I k } k is ultra log-concave (Mason’s conjecture) Cynthia Vinzant Completely log-concave polynomials and matroids
Other results Theorem: For any matroid M , the solution to the concave program n � p i log 1 1 τ = max p i + (1 − p i ) log 1 − p i p ∈P M i =1 can be computed in polynomial time and β = e τ satisfies 2 O ( − r ) β ≤ # bases of M ≤ β. Cynthia Vinzant Completely log-concave polynomials and matroids
Other results Theorem: For any matroid M , the solution to the concave program n � p i log 1 1 τ = max p i + (1 − p i ) log 1 − p i p ∈P M i =1 can be computed in polynomial time and β = e τ satisfies 2 O ( − r ) β ≤ # bases of M ≤ β. Theorem: The natural Markov Chain P ( B , B ′ ) on the bases of any rank- r matroid on [ n ] mixes quickly: min { t ∈ N : || P t ( B , · ) − π || 1 ≤ ǫ } ≤ r 2 log( n /ǫ ) . Cynthia Vinzant Completely log-concave polynomials and matroids
Sum up: completely log-concave polynomials ◮ log-concavity of polynomial as functions ⇒ log-concavity of coefficients ◮ many matroid polynomials are completely log-concave ◮ the theory of stable polynomials extends to CLC polynomials ⇒ Cynthia Vinzant Completely log-concave polynomials and matroids
Sum up: completely log-concave polynomials ◮ log-concavity of polynomial as functions ⇒ log-concavity of coefficients ◮ many matroid polynomials are completely log-concave ◮ the theory of stable polynomials extends to CLC polynomials ⇒ Thanks! Cynthia Vinzant Completely log-concave polynomials and matroids
References ◮ Karim Adiprasito, June Huh, Eric Katz, Hodge theory for combinatorial geometries , Annals of Mathematics 188(2), 2018. ◮ Nima Anari, Shayan Oveis Gharan, Cynthia Vinzant, Log-Concave Polynomials I: Entropy and a Deterministic Approximation Algorithm for Counting Bases of Matroids , arXiv:1807.00929 ◮ Nima Anari, KuiKui Liu, Shayan Oveis Gharan, Cynthia Vinzant, Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid , arXiv:1811.01816 ◮ Nima Anari, KuiKui Liu, Shayan Oveis Gharan, Cynthia Vinzant, Log-Concave Polynomials III: Mason’s Ultra-Log-Concavity Conjecture for Independent Sets of Matroids , arXiv:1811.01600 ◮ Petter Br¨ and´ en, Polynomials with the half-plane property and matroid theory , Advances in Mathematics 216(1), 2007. ◮ Petter Br¨ and´ en, June Huh, Hodge-Riemann relations for Potts model partition functions , arXiv:1811.01696 ◮ Young-Bin Choe, James Oxley, Alan Sokal, David Wagner, Homogeneous multivariate polynomials with the half-plane property , Advances in Applied Mathematics, 32(1-2), 2004. ◮ Leonid Gurvits, On multivariate Newton-like inequalities , Advances in combinatorial mathematics, 61–78, 2009. ◮ June Huh, Benjamin Schr¨ oter, Botong Wang, Correlation bounds for fields and matroids , arXiv:1806.02675 Cynthia Vinzant Completely log-concave polynomials and matroids
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