In
mathematics, Thomae's formula is a formula introduced by relating
theta constant In mathematics, a theta constant or
Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction ''θ'm''(''τ'') = θ''m''(''τ'',''0'') of a theta function ''θ'm''(τ,''z'') with rational characteristic ...
s to the
branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...
s of a
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
.
History
In 1824 the
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
established that
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
s 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
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermi ...
,
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
, and
Francesco Brioschi
Francesco Brioschi (22 December 1824 – 13 December 1897) was an Italian mathematician.
Biography
Brioschi was born in Milan in 1824. He graduated from the Collegio Borromeo in 1847.
From 1850 he taught analytical mechanics in the University ...
independently discovered that the
quintic equation
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
could be solved with
elliptic transcendents. This proved to be a generalization of the radical, which can be written as:
With the restriction to only this exponential, as shown by
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, only compositions of
Abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...
s may be constructed, which suffices only for equations of the fourth degree and below. Something more general is required for equations of higher degree, so to solve the quintic, Hermite, et al. replaced the exponential by an
elliptic modular function
In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is hol ...
and the integral (logarithm) by an
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
. Kronecker believed that this was a special case of a still more general method.
[
] Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
showed
[
] that any algebraic equation may be solved by use of modular functions. This was accomplished by Thomae in 1870.
Thomae generalized Hermite's approach by replacing the elliptic modular function with even more general
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...
s and the elliptic integral by a
hyperelliptic integral
In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1 ...
. Hiroshi Umemura
[
] expressed these modular functions in terms of higher genus
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
s.
Formula
If we have a
polynomial function:
with
irreducible over a certain subfield of the complex numbers, then its roots
may be expressed by the following equation involving
theta functions
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum fiel ...
of zero argument (
theta constant In mathematics, a theta constant or
Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction ''θ'm''(''τ'') = θ''m''(''τ'',''0'') of a theta function ''θ'm''(τ,''z'') with rational characteristic ...
s):
where
is the
period matrix In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures.
Ehresmann's theorem
Let be a holomorphic submersive morphism. For a point ''b'' of ''B'', we deno ...
derived from one of the following hyperelliptic integrals. If
is of odd degree, then,
Or if
is of even degree, then,
This formula applies to any algebraic equation of any degree without need for a
Tschirnhaus transformation
In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.
Simply, it is a method for transforming a polynomial equation ...
or any other manipulation to bring the equation into a specific normal form, such as the
Bring–Jerrard form
In algebra, the Bring radical or ultraradical of a real number ''a'' is the unique real root of the polynomial
: x^5 + x + a.
The Bring radical of a complex number ''a'' is either any of the five roots of the above polynomial (it is thus ...
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
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*
{{refend
Riemann surfaces
Theorems in number theory
Polynomials
Equations
Modular forms