Belyi Maps from Zeroes of Hypergeometric Polynomials
The evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation <inline-formula>&l...
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2025-01-01
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| author | Raimundas Vidunas |
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| description | The evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi mathvariant="normal">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mrow><mo>(</mo><mo>−</mo><mi>N</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>;</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps. Thereby, this article contributes to the systematic enumeration of Belyi maps. |
| format | Article |
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| institution | Kabale University |
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| spelling | doaj-art-fb94148efcc845ffa002dd40a76d59862025-01-10T13:18:26ZengMDPI AGMathematics2227-73902025-01-0113115610.3390/math13010156Belyi Maps from Zeroes of Hypergeometric PolynomialsRaimundas Vidunas0Institute of Applied Mathematics, Vilnius University, 03225 Vilnius, LithuaniaThe evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi mathvariant="normal">F</mi><mn>1</mn><none></none><mprescripts></mprescripts><mn>2</mn><none></none></mmultiscripts><mrow><mo>(</mo><mo>−</mo><mi>N</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>;</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula>. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps. Thereby, this article contributes to the systematic enumeration of Belyi maps.https://www.mdpi.com/2227-7390/13/1/156Gauss hypergeometric functionBelyi map (of genus 0)elliptic surfaces |
| spellingShingle | Raimundas Vidunas Belyi Maps from Zeroes of Hypergeometric Polynomials Mathematics Gauss hypergeometric function Belyi map (of genus 0) elliptic surfaces |
| title | Belyi Maps from Zeroes of Hypergeometric Polynomials |
| title_full | Belyi Maps from Zeroes of Hypergeometric Polynomials |
| title_fullStr | Belyi Maps from Zeroes of Hypergeometric Polynomials |
| title_full_unstemmed | Belyi Maps from Zeroes of Hypergeometric Polynomials |
| title_short | Belyi Maps from Zeroes of Hypergeometric Polynomials |
| title_sort | belyi maps from zeroes of hypergeometric polynomials |
| topic | Gauss hypergeometric function Belyi map (of genus 0) elliptic surfaces |
| url | https://www.mdpi.com/2227-7390/13/1/156 |
| work_keys_str_mv | AT raimundasvidunas belyimapsfromzeroesofhypergeometricpolynomials |