Acta Mathematica

Complete monotonicity for inverse powers of some combinatorially defined polynomials

Alexander D. Scott and Alan D. Sokal

Full-text: Open access

Abstract

We prove the complete monotonicity on (0,)n for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab-initio methods for proving that P-β is completely monotone on a convex cone C: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of P-β for some β>0 can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for P, and is also related to the Rayleigh property for matroids.

Article information

Source
Acta Math., Volume 213, Number 2 (2014), 323-392.

Dates
Received: 11 January 2013
Revised: 13 November 2013
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485801868

Digital Object Identifier
doi:10.1007/s11511-014-0121-6

Mathematical Reviews number (MathSciNet)
MR3286037

Zentralblatt MATH identifier
1304.05074

Subjects
Primary: 05C31 (Primary)
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A20: Combinatorial inequalities 05B35: Matroids, geometric lattices [See also 52B40, 90C27] 05C05: Trees 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 05E99: None of the above, but in this section 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 15B33: Matrices over special rings (quaternions, finite fields, etc.) 15B57: Hermitian, skew-Hermitian, and related matrices 17C99: None of the above, but in this section 26A48: Monotonic functions, generalizations 26B25: Convexity, generalizations 26C05: Polynomials: analytic properties, etc. [See also 12Dxx, 12Exx] 32A99: None of the above, but in this section 43A85: Analysis on homogeneous spaces 44A10: Laplace transform 60C05: Combinatorial probability 82B20 (Secondary)

Keywords
complete monotonicity positivity inverse power fractional power polynomial spanning-tree polynomial basis generating polynomial elementary symmetric polynomial matrixtree theorem determinant quadratic form half-plane property Hurwitz stability Rayleigh property Bernstein–Hausdorff–Widder theorem Laplace transform harmonic analysis symmetric cone Euclidean Jordan algebra Gindikin–Wallach set

Rights
2014 © Institut Mittag-Leffler

Citation

Scott, Alexander D.; Sokal, Alan D. Complete monotonicity for inverse powers of some combinatorially defined polynomials. Acta Math. 213 (2014), no. 2, 323--392. doi:10.1007/s11511-014-0121-6. https://projecteuclid.org/euclid.acta/1485801868


Export citation

References

  • Abdesselam, A. (2004) The Grassmann–Berezin calculus and theorems of the matrix-tree type. Adv. in Appl. Math., 33, 51–70.
  • Aczél, J., Lectures on Functional Equations and their Applications. Mathematics in Science and Engineering, 19. Academic Press, New York–London, 1966.
  • Aitken, A., Determinants and Matrices, 9th edition. Oliver and Boyd, Edinburgh, 1956.
  • Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis. Hafner, New York, 1965.
  • Anderson, T. W., An Introduction to Multivariate Statistical Analysis. Wiley Series in Probability and Statistics. Wiley-Interscience, Hoboken, NJ, 2003.
  • Askey, R., Summability of Jacobi series. Trans. Amer. Math. Soc., 179 (1973), 71–84.
  • Askey, R., Orthogonal Polynomials and Special Functions. Soc. Ind. Appl. Math., Philadelphia, PA, 1975.
  • Askey, R. & Gasper, G., Certain rational functions whose power series have positive coefficients. Amer. Math. Monthly, 79 (1972), 327–341.
  • Askey, R. & Gasper, G., Convolution structures for Laguerre polynomials. J. Anal. Math., 31 (1977), 48–68.
  • Askey, R. & Pollard, H., Some absolutely monotonic and completely monotonic functions. SIAM J. Math. Anal., 5 (1974), 58–63.
  • Atanasiu, D., Laplace integral on rational numbers. Math. Scand., 76 (1995), 152–160.
  • Baclawski, K. & White, N. L., Higher order independence in matroids. J. London Math. Soc., 19 (1979), 193–202.
  • Baez, J. C., The octonions. Bull. Amer. Math. Soc., 39 (2002), 145–205; errata 42 (2005), 213.
  • Bapat, R. B. & Raghavan, T. E. S., Nonnegative Matrices and Applications. Encyclopedia of Mathematics and its Applications, 64. Cambridge Univ. Press, Cambridge, 1997.
  • Baryshnikov, Y. & Pemantle, R., Asymptotics of multivariate sequences, part III: Quadratic points. Adv. Math., 228 (2011), 3127–3206.
  • Berezin, F. A., Quantization in complex symmetric spaces. Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 363–402, 472 (Russian); English translation in Math. USSR–Izv., 9 (1975), 341–379.
  • Berg, C., Stieltjes–Pick–Bernstein–Schoenberg and their connection to complete monotonicity, in Positive Definite Functions: From Schoenberg to Space-Time Challenges. Department of Mathematics, Universitat Jaume I de Castelló, Castelló, 2008. Also available at
  • Berg, C., Christensen, J.P. R. & Ressel, P., Harmonic Analysis on Semigroups. Graduate Texts in Mathematics, 100. Springer, New York, 1984.
  • Berntein, J. N., Analytic continuation of generalized functions with respect to a parameter. Funktsional. Anal. i Prilozhen., 6 (1972), 26–40 (Russian); English translation in Funct. Anal. Appl., 6 (1972), 273–285.
  • Björk, J.-E., Rings of Differential Operators. North-Holland Mathematical Library, 21. North-Holland, Amsterdam–New York, 1979.
  • Blekher, P. M., Integration of functions in a space of complex dimensions. Teoret. Mat. Fiz., 50 (1982), 370–382 (Russian); English translation in Theoret. and Math. Phys., 50 (1982), 243–251.
  • Bonnefoy-Casalis, M., Familles exponentielles naturelles invariantes par un groupe. Ph.D. Thesis, Laboratoire de Statistique et Probabilités, Université Paul Sabatier, Toulouse, 1990.
  • Borcea, J., Brändén, P. & Liggett, T. M., Negative dependence and the geometry of polynomials. J. Amer. Math. Soc., 22 (2009), 521–567.
  • Brändén, P., Polynomials with the half-plane property and matroid theory. Adv. Math., 216 (2007), 302–320.
  • Brändén, P., Solutions to two problems on permanents. Linear Algebra Appl., 436 (2012), 53–58.
  • Brändén, P. & González D’León, R. S., On the half-plane property and the Tutte group of a matroid. J. Combin. Theory Ser. B, 100 (2010), 485–492.
  • Brandstädt, A., Le, V. B. & Spinrad, J. P., Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. Soc. Ind. Appl. Math., Philadelphia, PA, 1999.
  • Brooks, R. L., Smith, C. A. B., Stone, A.H. & Tutte, W. T., The dissection of rectangles into squares. Duke Math. J., 7 (1940), 312–340.
  • Caracciolo, S., Sokal, A. D. & Sportiello, A., Grassmann integral representation for spanning hyperforests. J. Phys. A, 40:46 (2007), 13799–13835.
  • Caracciolo, S., Sokal, A. D. & Sportiello, A., Analytic continuation in dimension and its supersymmetric extension. In preparation.
  • Casalis, M. & Letac, G., Characterization of the Jørgensen set in generalized linear models. TEST, 3 (1994), 145–162.
  • Casalis, M. & Letac, G., The Lukacs–Olkin–Rubin characterization of Wishart distributions on symmetric cones. Ann. Statist., 24 (1996), 763–786.
  • Chaiken, S., A combinatorial proof of the all minors matrix tree theorem. SIAM J. Algebraic Discrete Methods, 3 (1982), 319–329.
  • Chaiken, S. & Kleitman, D. J., Matrix tree theorems. J. Combin. Theory Ser. A, 24 (1978), 377–381.
  • Chen, W.-K., Applied Graph Theory. North-Holland Series in Applied Mathematics and Mechanics, 13. North-Holland, Amsterdam–New York–Oxford, 1976.
  • Choe, Y.-B., Polynomials with the half-plane property and the support theorems. J. Combin. Theory Ser. B, 94 (2005), 117–145.
  • Choe, Y.-B., Oxley, J. G., Sokal, A. D. & Wagner, D. G., Homogeneous multivariate polynomials with the half-plane property. Adv. in Appl. Math., 32 (2004), 88–187.
  • Choe, Y.-B. &Wagner, D. G., Rayleigh matroids. Combin. Probab. Comput., 15 (2006), 765–781.
  • Choquet, G., Deux exemples classiques de représentation intégrale. Enseign. Math., 15 (1969), 63–75.
  • Colbourn, C. J., The Combinatorics of Network Reliability. International Series of Monographs on Computer Science. Oxford Univ. Press, New York, 1987.
  • Coxeter, H. S.M. & Greitzer, S. L., Geometry Revisited. New Mathematical Library, 19. Random House, New York, 1967.
  • Devinatz, A., The representation of functions as a Laplace–Stieltjes integrals. Duke Math. J., 22 (1955), 185–191.
  • Devinatz, A. & Nussbaum, A. E., Real characters of certain semi-groups with applications. Duke Math. J., 28 (1961), 221–237.
  • Diestel, R., Graph Theory. Graduate Texts in Mathematics, 173. Springer, Heidelberg, 2010.
  • Dray, T. & Manogue, C. A., The octonionic eigenvalue problem. Adv. Appl. Clifford Algebras, 8 (1998), 341–364.
  • Duffin, R. J., Topology of series-parallel networks. J. Math. Anal. Appl., 10 (1965), 303–318.
  • Duistermaat, J. J., M. Riesz’s families of operators. Nieuw Arch. Wisk., 9 (1991), 93– 101.
  • Erdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G., Tables of Integral Transforms. Vol. I. McGraw-Hill, New York–Toronto–London, 1954.
  • Etingof, P., Note on dimensional regularization, in Quantum Fields and Strings: A Course for Mathematicians (Princeton, NJ, 1996/1997), Vol. 1, pp. 597–607. Amer. Math. Soc., Providence, RI, 1999.
  • Faraut, J., Formule du binôme généralisée, in Harmonic Analysis (Luxembourg, 1987), Lecture Notes in Math., 1359, pp. 170–180. Springer, Berlin–Heidelberg, 1988.
  • Faraut, J. & Korányi, A., Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal., 88 (1990), 64–89.
  • Faraut, J. & Korányi, A., Analysis on Symmetric Cones. Oxford Mathematical Monographs. Oxford Univ. Press, New York, 1994.
  • Feder, T. & Mihail, M., Balanced matroids, in Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing (Victoria, DC, 1992), pp. 26–38. ACM, New York, 1992.
  • Fields, J. L. & Ismail, M.E. H., On the positivity of some 1F2’s. SIAM J. Math. Anal., 6 (1975), 551–559.
  • Freudenthal, H., Beziehungen der E7 und E8 zur Oktavenebene. I. Indag. Math., 16 (1954), 218–230.
  • Gasper, G., Positive integrals of Bessel functions. SIAM J. Math. Anal., 6 (1975), 868– 881.
  • Gillis, J., Reznick, B. & Zeilberger, D., On elementary methods in positivity theory. SIAM J. Math. Anal., 14 (1983), 396–398.
  • Gindikin, S.G., Invariant generalized functions in homogeneous domains. Funktsional. Anal. i Prilozhen., 9 (1975), 56–58 (Russian); English translation in Funct. Anal. Appl., 9 (1975), 50–52.
  • Glöckner, H., Positive definite functions on infinite-dimensional convex cones. Mem. Amer. Math. Soc., 166:789 (2003).
  • Goodman, N. R., Statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Statist., 34 (1963), 152–177.
  • Graczyk, P., Letac, G. & Massam, H., The complex Wishart distribution and the symmetric group. Ann. Statist., 31 (2003), 287–309.
  • Gurau, R., Magnen, J. & Rivasseau, V., Tree quantum field theory. Ann. Henri Poincaré, 10 (2009), 867–891.
  • Hilgert, J. & Neeb, K.H., Vector valued Riesz distributions on Euclidian Jordan algebras. J. Geom. Anal., 11 (2001), 43–75.
  • Hirsch, F., Familles résolvantes générateurs, cogénérateurs, potentiels. Ann. Inst. Fourier (Grenoble), 22 (1972), 89–210.
  • Hörmander, L., The Analysis of Linear Partial Differential Operators. I. Grundlehren der Mathematischen Wissenschaften, 256. Springer, Berlin–Heidelberg, 1990.
  • Horn, R. A., On infinitely divisible matrices, kernels, and functions. Z. Wahrsch. Verw. Gebiete, 8 (1967), 219–230.
  • Horn, R. A., The theory of infinitely divisible matrices and kernels. Trans. Amer. Math. Soc., 136 (1969), 269–286.
  • Horn, R. A., Infinitely divisible positive definite sequences. Trans. Amer. Math. Soc., 136 (1969), 287–303.
  • Ingham, A. E., An which occurs in statistics. Proc. Camb. Philos. Soc., 29 (1933), 271– 276.
  • Ishi, H., Positive Riesz distributions on homogeneous cones. J. Math. Soc. Japan, 52 (2000), 161–186.
  • Ismail, M. E.H. & Tamhankar, M. V., A combinatorial approach to some positivity problems. SIAM J. Math. Anal., 10 (1979), 478–485.
  • Kaluza, T., Elementarer Beweis einer Vermutung von K. Friedrichs und H. Lewy. Math. Z., 37 (1933), 689–697.
  • Kauers, M. & Zeilberger, D., Experiments with a positivity-preserving operator. Experiment. Math., 17 (2008), 341–345.
  • Kirchhoff, G., Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme gefürht wird. Ann. Phys., 148 (1847), 497–508.
  • Koornwinder, T., Positivity proofs for linearization and connection coefficients of orthogonal polynomials satisfying an addition formula. J. London Math. Soc., 18 (1978), 101–114.
  • Korepin, V. E., Bogoliubov, N. M. & Izergin, A.G., Quantum Inverse Scattering Method and Correlation Functions. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, 1993.
  • Lassalle, M., Algèbre de Jordan et ensemble de Wallach. Invent. Math., 89 (1987), 375–393.
  • Marcus, M., Finite Dimensional Multilinear Algebra. Part II. Pure and Applied Mathematics, 23. Marcel Dekker, New York, 1975.
  • Massam, H., An exact decomposition theorem and a unified view of some related distributions for a class of exponential transformation models on symmetric cones. Ann. Statist., 22 (1994), 369–394.
  • Massam, H. & Neher, E., On transformations and determinants of Wishart variables on symmetric cones. J. Theoret. Probab., 10 (1997), 867–902.
  • Moak, D. S., Completely monotonic functions of the form s-b(s2+1)-a. Rocky Mountain J. Math., 17 (1987), 719–725.
  • Moldovan, M. M. & Gowda, M. S., Strict diagonal dominance and a Geršgorin type theorem in Euclidean Jordan algebras. Linear Algebra Appl., 431 (2009), 148–161.
  • Moon, J. W., Counting Labelled Trees. Canadian Mathematical Congress, Montreal, QC, 1970.
  • Moon, J. W., Some determinant expansions and the matrix-tree theorem. Discrete Math., 124 (1994), 163–171.
  • Muirhead, R. J., Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 1982.
  • Nerode, A. & Shank, H., An algebraic proof of Kirchhoff’s network theorem. Amer. Math. Monthly, 68 (1961), 244–247.
  • Nussbaum, A.E., The Hausdorff–Bernstein–Widder theorem for semi-groups in locally compact Abelian groups. Duke Math. J., 22 (1955), 573–582.
  • Oxley, J. G., Graphs and series-parallel networks, in Theory of Matroids, Encyclopedia Math. Appl., 26, pp. 97–126. Cambridge Univ. Press, Cambridge, 1986.
  • Oxley, J. G., Private communication. March 2008 and September 2011.
  • Oxley, J. G., Matroid Theory. Oxford Graduate Texts in Mathematics, 21. Oxford Univ. Press, Oxford, 2011.
  • Oxley, J.G., Vertigan, D. & Whittle, G., On maximum-sized near-regular and 16-matroids. Graphs Combin., 14 (1998), 163–179.
  • Pemantle, R., Analytic combinatorics in d variables: an overview, in Algorithmic Probability and Combinatorics, Contemp. Math., 520, pp. 195–220. Amer. Math. Soc., Providence, RI, 2010.
  • Pemantle, R. & Wilson, M. C., Asymptotics of multivariate sequences. I. Smooth points of the singular variety. J. Combin. Theory Ser. A, 97 (2002), 129–161.
  • Pemantle, R. & Wilson, M. C., Asymptotics of multivariate sequences. II. Multiple points of the singular variety. Combin. Probab. Comput., 13 (2004), 735–761.
  • Pemantle, R. & Wilson, M. C., Twenty combinatorial examples of asymptotics derived from multivariate generating functions. SIAM Rev., 50 (2008), 199–272.
  • Pendavingh, R. A. & van Zwam, S. H. M., Skew partial fields, multilinear representations of matroids, and a matrix tree theorem. Adv. in Appl. Math., 50 (2013), 201–227.
  • Prells, U., Friswell, M. I. & Garvey, S.D., Use of geometric algebra: compound matrices and the determinant of the sum of two matrices. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 273–285.
  • Reed, M. & Simon, B., Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness. Academic Press, New York–London, 1975.
  • Riesz, M., L’intégrale de Riemann–Liouville et le problème de Cauchy. Acta Math., 81 (1949), 1–223.
  • Robertson, N. & Seymour, P. D., Graph minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B, 92 (2004), 325–357.
  • Rossi, H. & Vergne, M., Analytic continuation of the holomorphic discrete series of a semi-simple Lie group. Acta Math., 136 (1976), 1–59.
  • Royle, G. & Sokal, A.D., The Brown–Colbourn conjecture on zeros of reliability polynomials is false. J. Combin. Theory Ser. B, 91 (2004), 345–360.
  • Rump, S. M., Theorems of Perron–Frobenius type for matrices without sign restrictions. Linear Algebra Appl., 266 (1997), 1–42.
  • Schwartz, L., Théorie des Distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Hermann, Paris, 1966.
  • Scott, A.D. & Sokal, A. D., Complete monotonicity for inverse powers of some combinatorially defined polynomials. Expanded version, 2013.
  • Shanbhag, D.N., The Davidson–Kendall problem and related results on the structure of the Wishart distribution. Austral. J. Statist., 30A (1988), 272–280.
  • Shucker, D. S., Extensions and generalizations of a theorem of Widder and of the theory of symmetric local semigroups. J. Funct. Anal., 58 (1984), 291–309.
  • Siegel, C. L., Über die analytische Theorie der quadratischen Formen. Ann. of Math., 36 (1935), 527–606.
  • Sokal, A.D., The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, in Surveys in Combinatorics 2005, London Math. Soc. Lecture Note Ser., 327, pp. 173–226. Cambridge Univ. Press, Cambridge, 2005.
  • Sokal, A.D., When is a Riesz distribution a complex measure? Bull. Soc. Math. France, 139 (2011), 519–534.
  • Speer, E. R., Dimensional and analytic renormalization, in Renormalization Theory (Erice, 1975), NATO Advanced Study Inst. Series C: Math. and Phys. Sci., 23, pp. 25–93. Reidel, Dordrecht, 1976.
  • Straub, A., Positivity of Szegő’s rational function. Adv. in Appl. Math., 41 (2008), 255–264.
  • Szegő, G., Über gewisse Potenzreihen mit lauter positiven Koeffizienten. Math. Z., 37 (1933), 674–688.
  • Terras, A., Harmonic Analysis on Symmetric Spaces and Applications. II. Springer, Berlin–Heidelberg, 1988.
  • Thomas, E. G. F., Bochner and Bernstein theorems via the nuclear integral representation theorem. J. Math. Anal. Appl., 297 (2004), 612–624.
  • Vere-Jones, D., A generalization of permanents and determinants. Linear Algebra Appl., 111 (1988), 119–124.
  • Wagner, D.G., Matroid inequalities from electrical network theory. Electron. J. Combin., 11 (2004/06), Article 1, 17 pp.
  • Wagner, D.G., Negatively correlated random variables and Mason’s conjecture for independent sets in matroids. Ann. Comb., 12 (2008), 211–239.
  • Wagner, D.G., Multivariate stable polynomials: theory and applications. Bull. Amer. Math. Soc., 48 (2011), 53–84.
  • Wagner, D.G. & Wei, Y., A criterion for the half-plane property. Discrete Math., 309:6 (2009), 1385–1390.
  • Wallach, N. R., The analytic continuation of the discrete series. II. Trans. Amer. Math. Soc., 251 (1979), 19–37.
  • Watson, G. N., A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge, 1944.
  • Whittle, G., On matroids representable over GF(3) and other fields. Trans. Amer. Math. Soc., 349 (1997), 579–603.
  • Widder, D.V., The Laplace Transform. Princeton Mathematical Series, 6. Princeton Univ. Press, Princeton, NJ, 1941.
  • Zastavnyi, V. P., On positive definiteness of some functions. J. Multivariate Anal., 73 (2000), 55–81.
  • Zeilberger, D., A combinatorial approach to matrix algebra. Discrete Math., 56 (1985), 61–72.