Anharmonic solutions to the Riccati equation and elliptic modular functions.
In: Forum Mathematicum, Jg. 30 (2018-03-01), Heft 2, S. 433-455
academicJournal
Zugriff:
We study the irreducible algebraic equation x n + a 1 x n - 1 + ⋯ + a n = 0 , with n ≥ 4 , x^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0,\quad\text{with ${n\geq 4}$,} on the differential field ( 𝔽 = ℂ ( t ) , δ = d d t ) {(\mathbb{F}=\mathbb{C}(t),\delta=\frac{d}{dt})}.We assume that a root of the equation is a solution to the Riccati differential equation u ′ + B 0 + B 1 u + B 2 u 2 = 0 {u^{\prime}+B_{0}+B_{1}u+B_{2}u^{2}=0} ,where B 0 {B_{0}} , B 1 {B_{1}} , B 2 {B_{2}} are in 𝔽 {\mathbb{F}}. We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial F n ( x , y ) ∈ ℂ ( x ) [ y ] {F_{n}(x,y)\in\mathbb{C}(x)[y]} such that for almost T ∈ 𝔽 ∖ ℂ {T\in\mathbb{F}\setminus\mathbb{C}} , the algebraic equation F n ( x , T ) = 0 {F_{n}(x,T)=0} is of the same type as the above stated algebraic equation.In other words, all its roots are solutions to the same Riccati equation.On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup Γ ( 2 ) {\Gamma(2)} , whose roots satisfy a common Riccati equation on the differential field ( ℂ ( E 2 , E 4 , E 6 ) , d d τ ) {(\mathbb{C}(E_{2},E_{4},E_{6}),\frac{d}{d\tau})} , with E i ( τ ) {E_{i}(\tau)} being the Eisenstein series of weight i .These solutions are related to a Darboux–Halphen system.Finally, we deal with the following problem: For which “potential” q ∈ ℂ ( ℘ , ℘ ′ ) {q\in\mathbb{C}(\wp,\wp^{\prime})} does the Riccati equation d Y d z + Y 2 = q {\frac{dY}{dz}+Y^{2}=q} admit algebraic solutions over the differential field ℂ ( ℘ , ℘ ′ ) {\mathbb{C}(\wp,\wp^{\prime})} , with ℘ {\wp} being the classical Weierstrass function?We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial Φ ( x ) {\Phi(x)} of an algebraic solution u must have a vanishing fourth transvectant τ 4 ( Φ ) ( x ) {\tau_{4}(\Phi)(x)}. [ABSTRACT FROM AUTHOR]
Titel: |
Anharmonic solutions to the Riccati equation and elliptic modular functions.
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Autor/in / Beteiligte Person: | Sebbar, Ahmed ; Wone, Oumar |
Zeitschrift: | Forum Mathematicum, Jg. 30 (2018-03-01), Heft 2, S. 433-455 |
Veröffentlichung: | 2018 |
Medientyp: | academicJournal |
ISSN: | 0933-7741 (print) |
DOI: | 10.1515/forum-2017-0025 |
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