Polynomial Solutions of Linear Differential Equations (Hardcover)
Language: English
Published by De Gruyter, Berlin, 2026
- Hardcover
- New
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Hardcover. Linear ordinary differential equations with polynomial coefficients and their solutions in the complex plane have been an active area of mathematical research since the mid-19th century, with contributions from such prominent figures as G. Lame, M. Bocher, E. Heine, F. Klein, T. Stieltjes, G. D. Birkhoff, and many others. During the 19th century, interest focused mainly on second-order equations because of their importance in physics. The existence of a polynomial solution to a linear differential equation is a very special property, implying in particular the existence of a one-dimensional invariant subspace under the action of the monodromy group on the space of solutions. Polynomial solutions rarely occur for individual linear differential equations, but they typically arise within families of such equations. In this book, we discuss in detail two major topics: exactly solvable linear differential operators, closely related to the BochnerKrall problem in orthogonal polynomials, and the (generalized) HeineStieltjes theory, both of which originated more than a century ago. Our main emphasis is on the study of the root asymptotics of various polynomial sequences appearing in both problems. We also establish connections between our asymptotic analysis and special classes of quadratic and higher-order differentials, as well as other objects of geometric origin. In addition, we present a wealth of numerical results and suggest many open problems for the interested reader. The Bibliotheca Teubneriana, established in 1849, has evolved into the world's most venerable and extensive series of editions of Greek and Latin literature, ranging from classical to Neo-Latin texts. Some 4-5 new editions are published every year. This item is printed on demand. Shipping may be from our Sydney, NSW warehouse or from our UK or US warehouse, depending on stock availability.…
Seller Inventory # 9783119147101
- Title
- Polynomial Solutions of Linear Differential Equations (Hardcover)
- Author
- Boris Shapiro
- Publisher
- De Gruyter, Berlin
- Publication year
- 2026
- Condition
- new
- Binding
- Hardcover
- Language
- English
- ISBN 10
- 3119147109
- ISBN 13
- 9783119147101
Linear ordinary differential equations with polynomial coefficients and their solutions in the complex plane have been an active area of mathematical research since the mid-19th century, with contributions from such prominent figures as G. Lamé, M. Bôcher, E. Heine, F. Klein, T. Stieltjes, G. D. Birkhoff, and many others. During the 19th century, interest focused mainly on second-order equations because of their importance in physics. The existence of a polynomial solution to a linear differential equation is a very special property, implying in particular the existence of a one-dimensional invariant subspace under the action of the monodromy group on the space of solutions. Polynomial solutions rarely occur for individual linear differential equations, but they typically arise within families of such equations.
In this book, we discuss in detail two major topics: exactly solvable linear differential operators, closely related to the Bochner–Krall problem in orthogonal polynomials, and the (generalized) Heine–Stieltjes theory, both of which originated more than a century ago. Our main emphasis is on the study of the root asymptotics of various polynomial sequences appearing in both problems. We also establish connections between our asymptotic analysis and special classes of quadratic and higher-order differentials, as well as other objects of geometric origin. In addition, we present a wealth of numerical results and suggest many open problems for the interested reader.
"Synopsis" may belong to another edition of this title.
About the Author
Prof. B.Shapiro has been working in Stockholm University since 1991. He has published around 130 articles in various areas of mathematics, mainly in differential equations, topology and commutative algebra. One of his (joint) achievements is the formulation of the so-called Shapiro-Shapiro conjecture in real Schubert calculus which has been later settled by Mukhin-Tarasov-Varchenko. Another is his finding of a very intriguing, but completely forgotten conjecture on the number of equilibrium points for systems of point charges in 3-dimensional space due to J.C.Maxwell which was published in his book “A Treatise on Electricity and Magnetism” in 1873.
Prof. B.Shapiro is a permanent recipient of research grants from the Swedish Science Foundation and served as an advisor for 12 doctoral students.
"About the title" may belong to another edition of this title.
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