The fundamental theorem of algebra states that every polynomial
equation of degree n has n roots in the
complex plane (counting multiplicity). The picture on the left
shows log[1 + q[z]] where q is a quintic.
The picture on the right shows the lines where the real and imaginary
parts of q[z] are zero; they cross at right angles at
the roots.
The general solution of the quadratic equation was found more than
4000 years ago. The solutions of the cubic and quarticfound in
the 1500swere major results of Renaissance mathematics.
Mathematicians struggled for centuries to find formulas for the
solutions of equations of higher degree, but despite the efforts
of Euler, Bezout, Malfatti,
Lagrange, and others, no general
solutions were found. Finally, Ruffini
(1799) and Abel (1826)
showed that the solution of the general quintic cannot be written
as a finite formula involving only the four arithmetic operations
and the extraction of roots. By 1832 Galois had developed the
theory of Galois groups and described exactly when a polynomial
equation is solvable.
The colored squares represent the Galois groups of the equation:
in the ab plane for integers a and
b . Gray squares represent quintics that factor
into lower degree polynomials. Orange and red squares represent
those quintics that don't factor but that can still be solved in
radicals. The dark and light blue squares correspond to quintics
that cannot be solved in radicals.
There are polynomials of degree greater than four which do not
factor over the rationals, but that can still be solved in radicals.
An example is the quintic:
one of whose roots is
In 1683 Tschirnhaus published
Method of Eliminating All Intermediate
Terms from a Given Equation. Although the title is exaggerated,
Tschirnhaus's paper was the most important idea for the solution of
algebraic equations in about 200 years.
Tschirnhaus's transformation reduces the n th degree
polynomial equation:
to one with up to three fewer terms:
by transforming the roots as follows:
where the gamma j can be expressed in radicals in terms of the
a j. Thus every quintic can be transformed into one of
the form:
The b j can ultimately be expressed in radicals in
terms of the a j. The resulting expressions are
typically enormous; for a general quintic with symbolic coefficients
they require megabytes of storage.
The group structure of the icosahedral equation is related to the
stereographic projection of the triangulated regular icosahedron.
In 1877 Klein published
Lectures on the Icosahedron and the Solution
of Equations of the Fifth Degree. In this remarkable book and
later article Klein gave a complete solution of the quintic.
He used a Tschirnhaus transformation to reduce the general quintic
to the form:
He found the solution of this reduced quintic by first solving the
related icosahedral equation:
where Z can be expressed in radicals in terms of
a , b , and c .
A solution of the icosahedral equation using
hypergeometric functions is:
Johann Lambert (1757) seems to have
been the first person to have the idea of using series to solve a
polynomial equation. In the next 150 years similar ideas were
raised independently by Leonard Euler
(1770), Pafnuti Chebyshev (1838),
and Gotthold Eisenstein (1844),
among others.
We first illustrate the idea with the
quadratic and then go on to the
quintic. Consider the generic quadratic polynomial:
Let r[y] be a branch of the inverse function of q
in a small neighborhood about y = 0 , meaning that
q[r[y]] = y for any sufficiently small y .
Then r[0] is clearly a root of q .
In order to find r[0] as a funtion of a ,
b , and c , we will compute the formal power
series of r[y] in y at zero; for this it
suffices to invert q[x] as a formal power series:
In[1]:=
Out[1]=
This gives nine terms of the series, which we evaluate at zero to
get r[0] :
In[2]:=
Out[2]=
The general term is:
In[3]:=
We can sum this series to get a closedform solution for one of the
roots of the original quadratic equation:
In[4]:=
Out[4]=
Now we show how this method works for a quintic of the form:
Proceeding as above, we compute:
In[5]:=
Out[5]=
This gives the first eight terms in the series for one of the roots.
The general term of this series can be written as:
In[6]:=
This series can be summed using hypergeometric
functions; one of the roots of the quintic is:
In[7]:=
Out[7]=
In principle this method will work for any polynomial.
Galois groups of quintics are related to the symmetries of the
icosahedron.
Cockle (1860) and Harley (1862)
developed a method for solving algebraic equations based on
differential equations.
We illustrate their approach with the quintic equation:
In[1]:=
Each root is a function of the paremeter rho . The
differential resolvent of this quintic is a linear differential
equation of order four of the form:
In[2]:=
The coefficients a j are polynomial functions
in rho . To find them explicitly, we differentiate
the quintic equation with respect to rho and substitute
these derivatives into the differential equation. This gives a
4 th degree algebraic equation in
t[rho] . Setting the coefficients of the powers of
t[rho] to zero gives a system of five linear
equations in the a j. Solving this system
gives the differential resolvent:
In[3]:=
We solve the differential equation Omega :
In[4]:=
This is a general solution to the differential equation that depends on
the four parameters c1 , c2 , c3 ,
and c4 . We will find the values of these parameters so that
this solution will also satisfy the original quintic. To do this we could
substitute this general solution into the quintic and solve the resulting
equation.
However, it is enough to substitute the series expansion of the solution
at rho = 0 , where the main term of the series expansion
of the generalized hypergeometric function is 1 :
In[5]:=
In[6]:=
Now we collect like terms in rho and set the coefficients
equal to zero. We solve the resulting system of four linear equations
for c j:
In[7]:=
In[8]:=
Finally we have obtained the five roots of the given quintic:
The approach just described can be used for polynomials of any degree.
However, only polynomials that can be written as trinomials of the
form:
have roots that can be represented in terms of hypergeometric
functions of one variable.
Some of the ideas described here can be generalized to
equations of higher degree. The basic ideas for solving the sextic
using Klein's approach to the quintic were worked out around 1900.
For algebraic equations beyond the sextic, the roots can be expressed
in terms of hypergeometric functions in several variables or in terms
of Siegel modular functions.
