Zeros of discriminants constructed from Hermite–Padé polynomials of an algebraic function and their relation to branch points

A. V. Komlov   ; R. V. Palvelev

Sbornik. Mathematics, Tome 215 (2024) no. 12, pp. 1633-1665

Résumé

Let $f_\infty$ be the germ at $\infty$ of some algebraic function $f$ of degree $m+1$. Let $Q_{n,j}$, $j=0,\dots,m$, be the Hermite–Padé polynomials of the first type of order $n\in\mathbb N$ constructed from the tuple of germs $[1, f_ \infty, f_\infty^2,\dots,f_\infty^m]$. We study the asymptotic properties of discriminants constructed from the Hermite–Padé polynomials in question, that is, the discriminants $D_n(z)$ of the polynomials $Q_{n,m}(z)w^m+Q_{n,m-1}(z)w^{m-1}+\dots+Q_{n,0}(z)$. We find their weak asymptotics, as well as the asymptotic behaviour of their ratio with the polynomial $Q_{n,m}^{2m-2}$. In addition, we refine the weak asymptotic formulae for $D_n$ at branch points of the original algebraic function $f$ and apply the results obtained to the problem of finding branch points of $f$ numerically on the basis of the prescribed germ $f_\infty$, which is used in applied problems. Bibliography: 49 titles.

Détail

Keywords: branch points, algebraic functions, weak asymptotics.
Mots-clés : Hermite–Padé polynomials, discriminants

Bibliographie

[1] A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein and S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131 | DOI | MR | Zbl

[2] A. I. Aptekarev and V. A. Kalyagin, “Asymptotic behavior of an $n$th degree root of polynomials of simultaneous orthogonality, and algebraic functions”, Akad. Nauk SSSR Inst. Prikl. Mat. Preprint, 1986, 60, 18 pp. (Russian) | MR

[3] A. I. Aptekarev and A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Russian Math. Surveys, 66:6 (2011), 1133–1199 | DOI | MR | Zbl

[4] A. I. Aptekarev, G. López Lagomasino and A. Martínez-Finkelshtein, “On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials”, Russian Math. Surveys, 72:3 (2017), 389–449 | DOI | MR | Zbl

[5] A. I. Aptekarev and V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite–Padé approximants”, Sb. Math., 201:2 (2010), 183–234 | DOI | MR | Zbl

[6] A. I. Aptekarev and D. N. Tulyakov, “Geometry of Hermite–Padé approximants for system of functions $\{f,f^2\}$ with three branch points”, Keldysh Institute preprints, 2012, 77, 25 pp.

[7] A. I. Aptekarev and D. N. Tulyakov, “Nuttall's Abelian integral on the Riemann surface of the cube root of a polynomial of degree 3”, Izv. Math., 80:6 (2016), 997–1034 | DOI | MR | Zbl

[8] A. I. Aptekarev and M. L. Yattselev, “Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials”, Acta Math., 215:2 (2015), 217–280 | DOI | MR | Zbl

[9] V. I. Buslaev, A. Martínez-Finkelshtein and S. P. Suetin, “Method of interior variations and existence of $S$-compact sets”, Proc. Steklov Inst. Math., 279 (2012), 25–51 | DOI | MR | Zbl

[10] Xuanhao Chang, E. O. Dobrolyubov and S. V. Krasnoshchekov, “Fundamental studies of vibrational resonance phenomena by multivalued resummation of the divergent Rayleigh–Schrödinger perturbation theory series: deciphering polyad structures of three $H_2{ }^{16}O$ isotopologues”, Phys. Chem. Chem. Phys., 24:11 (2022), 6655–6675 | DOI

[11] E. M. Chirka, “Potentials on a compact Riemann surface”, Proc. Steklov Inst. Math., 301 (2018), 272–303 | DOI | MR | Zbl

[12] E. M. Chirka, “Riemann surfaces”, Lektsionnye Kursy NOTs, 1, Steklov Math. Institute of RAS, Moscow, 2006, 3–105 (Russian) | DOI | Zbl

[13] E. O. Dobrolyubov, N. R. Ikonomov, L. A. Knizhnerman and S. P. Suetin, Rational Hermite–Padé approximants vs Padé approximants. Numerical results, arXiv: 2306.07063

[14] E. O. Dobrolyubov, I. V. Polyakov, D. V. Millionshchikov and S. V. Krasnoshchekov, “Vibrational resonance phenomena of the OCS isotopologues studied by resummation of high-order Rayleigh–Schrödinger perturbation theory”, J. Quant. Spectrosc. Radiat. Transf., 316 (2024), 108909, 13 pp. | DOI

[15] A. N. Duchko and A. D. Bykov, “Multivalued property of Rayleigh–Schrödinger perturbation series for vibrational energy levels of molecules”, Phys. Scr., 94:10 (2019), 105403, 13 pp. | DOI

[16] A. A. Gonchar and E. A. Rakhmanov, “On convergence of simultaneous Padé approximants for systems of functions of Markov type”, Proc. Steklov Inst. Math., 157 (1983), 31–50 | MR | Zbl

[17] A. A. Gonchar, E. A. Rakhmanov and V. N. Sorokin, “Hermite–Padé approximants for systems of Markov-type functions”, Sb. Math., 188:5 (1997), 671–696 | DOI | MR | Zbl

[18] P. Henrici, “An algorithm for analytic continuation”, SIAM J. Numer. Anal., 3:1 (1966), 67–78 | DOI | MR | Zbl

[19] A. Katz, “The analytic structure of many-body perturbation theory”, Nuclear Phys., 29 (1962), 353–372 | DOI | MR

[20] R. K. Kovacheva and S. P. Suetin, “Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser”, Proc. Steklov Inst. Math., 284 (2014), 168–191 | DOI | MR | Zbl

[21] A. V. Komlov, “Polynomial Hermite–Padé $m$-system and reconstruction of the values of algebraic functions”, Extended abstracts fall 2019–spaces of analytic functions: approximation, interpolation, sampling, Trends Math. Res. Perspect. CRM Barc., 12, Birkhäuser/Springer, Cham, 2021, 113–121 | DOI | MR | Zbl

[22] A. V. Komlov, “The polynomial Hermite–Padé $m$-system for meromorphic functions on a compact Riemann surface”, Sb. Math., 212:12 (2021), 1694–1729 | DOI | MR | Zbl

[23] A. V. Komlov, N. G. Kruzhilin, R. V. Palvelev and S. P. Suetin, “Convergence of Shafer quadratic approximants”, Russian Math. Surveys, 71:2 (2016), 373–375 | DOI | MR | Zbl

[24] A. V. Komlov, R. V. Palvelev, S. P. Suetin and E. M. Chirka, “Hermite–Padé approximants for meromorphic functions on a compact Riemann surface”, Russian Math. Surveys, 72:4 (2017), 671–706 | DOI | MR | Zbl

[25] S. V. Krasnoshchekov, E. O. Dobrolyubov, M. A. Syzgantseva and R. V. Palvelev, “Rigorous vibrational Fermi resonance criterion revealed: two different approaches yield the same result”, Molec. Phys., 118:11 (2020), e1743887, 7 pp. | DOI

[26] S. Lvovski, Principles of Complex Analysis, Moscow Lectures, 6, Springer, Cham, 2020, xiii+257 pp.

[27] V. G. Lysov, “On Hermite–Padé approximants for the product of two logarithms”, Keldysh Institute preprints, 2017, 141, 24 pp. (Russian)

[28] V. G. Lysov, “Strong asymptotics of Hermite–Pade approximants for the Nikishin system of Jacobi weights”, Keldysh Institute preprints, 2017, 85, 35 pp. (Russian)

[29] J. Nuttall, “Hermite–Padé approximants to functions meromorphic on a Riemann surface”, J. Approx. Theory, 32:3 (1981), 233–240 | DOI | MR | Zbl

[30] J. Nuttall, “Asymptotics of diagonal Hermite–Padé polynomials”, J. Approx. Theory, 42:4 (1984), 299–386 | DOI | MR | Zbl

[31] E. A. Perevoznikova and E. A. Rakhmanov, Variation of the equilibrium energy and the $S$-property of compact sets of minimum capacity, preprint, Moscow, 1994 (Russian)

[32] V. V. Prasolov, Problems and theorems in linear algebra, Transl. Math. Monogr., 134, Amer. Math. Soc., Providence, RI, 1994, xviii+225 pp. | DOI | MR | Zbl

[33] E. A. Rakhmanov, “The asymptotics of Hermite–Padé polynomials for two Markov-type functions”, Sb. Math., 202:1 (2011), 127–134 | DOI | MR | Zbl

[34] E. A. Rakhmanov, “Zero distribution for Angelesco Hermite–Padé polynomials”, Russian Math. Surveys, 73:3 (2018), 457–518 | DOI | MR | Zbl

[35] E. A. Rakhmanov and S. P. Suetin, “The distribution of the zeros of the Hermite–Padé polynomials for a pair of functions forming a Nikishin system”, Sb. Math., 204:9 (2013), 1347–1390 | DOI | MR | Zbl

[36] T. Ransford, Potential theory in the complex plane, London Math. Soc. Stud. Texts, 28, Cambridge Univ. Press, Cambridge, 1995, x+232 pp. | DOI | MR | Zbl

[37] A. V. Sergeev and D. Z. Goodson, “Summation of asymptotic expansions of multiple-valued functions using algebraic approximants: application to anharmonic oscillators”, J. Phys. A, 31:18 (1998), 4301–4317 | DOI | Zbl

[38] A. V. Sergeev and D. Z. Goodson, “Singularities of Møller–Plesset energy functions”, J. Chem. Phys., 124:9 (2006), 094111, 11 pp. | DOI

[39] H. Stahl, “Extremal domains associated with an analytic function. I”, Complex Variables Theory Appl., 4:4 (1985), 311–324 | DOI | MR | Zbl

[40] H. Stahl, “Extremal domains associated with an analytic function. II”, Complex Variables Theory Appl., 4:4 (1985), 325–338 | DOI | MR | Zbl

[41] H. Stahl, “The structure of extremal domains associated with an analytic function”, Complex Variables Theory Appl., 4:4 (1985), 339–354 | DOI | MR | Zbl

[42] H. Stahl, “The convergence of Padé approximants to functions with branch points”, J. Approx. Theory, 91:2 (1997), 139–204 | DOI | MR | Zbl

[43] S. P. Suetin, Hermite–Padé polynomials and analytic continuation: new approach and some results, arXiv: 1806.08735

[44] S. P. Suetin, Maximum principle and asymptotic properties of Hermite–Padé polynomials, arXiv: 2109.10144

[45] S. P. Suetin, “Asymptotic properties of Hermite–Padé polynomials and Katz points”, Russian Math. Surveys, 77:6 (2022), 1149–1151 | DOI | MR | Zbl

[46] S. P. Suetin, “Existence of a three-sheeted Nutall surface for a certain class of infinite-valued analytic functions”, Russian Math. Surveys, 74:2 (2019), 363–365 | DOI | MR | Zbl

[47] S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padé polynomials for a Nikishin system”, Proc. Steklov Inst. Math., 301 (2018), 245–261 | DOI | MR | Zbl

[48] S. P. Suetin, “Scalar approaches to the limit distribution of zeros of Hermite–Padé polynomials for a Nikishin system”, Uspekhi Mat. Nauk, 80:1 (2025), 85–152 (Russian)

[49] S. P. Suetin, “Maximum principle and asymptotic properties of Hermite–Padé polynomials”, Russian Math. Surveys, 79:3 (2024), 547–549 | DOI | MR