In mathematics, Thomae's formula is a formula introduced by Carl Johannes Thomae (1870) relating theta constants to the branch points of a hyperelliptic curve (Mumford 1984, section 8).
History edit
In 1824 the Abel–Ruffini theorem established that polynomial equations of a degree of five or higher could have no solutions in radicals. It became clear to mathematicians since then that one needed to go beyond radicals in order to express the solutions to equations of the fifth and higher degrees. In 1858, Charles Hermite, Leopold Kronecker, and Francesco Brioschi independently discovered that the quintic equation could be solved with elliptic transcendents. This proved to be a generalization of the radical, which can be written as:
Formula edit
If we have a polynomial function:
This formula applies to any algebraic equation of any degree without need for a Tschirnhaus transformation or any other manipulation to bring the equation into a specific normal form, such as the Bring–Jerrard form for the quintic. However, application of this formula in practice is difficult because the relevant hyperelliptic integrals and higher genus theta functions are very complex.
References edit
- ^ Kronecker, Leopold (1858). "Sur la résolution de l'equation du cinquème degré". Comptes rendus de l'Académie des Sciences. 46: 1150–1152.
- ^ Jordan, Camille (1870). Traité des substitutions et des équations algébriques. Paris: Gauthier-Villars.
- ^ Thomae, Carl Johannes (1870). "Beitrag zur Bestimmung von θ(0,0,...0) durch die Klassenmoduln algebraischer Funktionen". Journal für die reine und angewandte Mathematik. 71: 201–222.
- ^ Umemura, Hiroshi (1984). "Resolution of algebraic equations by theta constants". In David Mumford (ed.). Tata Lectures on Theta II. Birkhäuser. pp. 3.261–3.272. ISBN 3-7643-3109-7.
- Mumford, David (1984), Tata lectures on theta. II, Progress in Mathematics, vol. 43, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3110-9, MR 0742776
- Thomae, Carl Johannes (1870), "Beitrag zur Bestimmung von θ(0,0,...0) durch die Klassenmoduln algebraischer Funktionen", Journal für die reine und angewandte Mathematik, 71: 201–222