Wednesday, October 29, 2008

Sturm's Theorem: An Initial Lemma

Here's a major problem that Jacques Charles Francois Sturm solved:

Find the number of real roots of an algebraic equation with real coefficients over a given interval.

There are two possible cases:

(I) The real roots of the equation in question are all simple over the given interval.

(II) The equation also possesses multiple real roots over the interval.

Lemma:

Any equation of multiple real roots can be broken down into a set of equations with only simple real roots.

Proof:

(1) Let the F(x) = 0 have the distinct roots α, β, γ, ...

(2) Let the root α be a-fold, the root β be b-fold, γ c-fold, etc where a,b,c,... are not necessarily 1.

(3) Using the Fundamental Theorem of Algebra (see Thereom, here), we have:

F(x) = (x - α)a(x - β)b(x - γ)c*...

(4) Using some basic properties of the derivative (see Corollary 2.2, here), we have:

F'(x)/F(x) = a/(x - α) + b/(x - β) + c/(x - γ) + ... =

= [a(x - β)(x - γ)(x - λ)*... + b(x - α)(x - γ)(x - λ)*... + ...]/[(x - α)(x - β)(x - γ)*...]

(5) Let:

p(x) = [a(x - β)(x - γ)(x - λ)*... + b(x - α)(x - γ)(x - λ)*... + ...]

q(x) = [(x - α)(x - β)(x - γ)*...]

so that we have:

F'(x)/F(x) = p(x)/q(x)

(6) We note that p(x) and q(x) do not have any common divisors since for each factor of q(x), we are left with a remainder of the form c/(x - d) where c,d are constants.

(7) Let G(x) = F(x)/q(x)

(8) Then:

F(x) = G(x)*q(x)

F'(x) = G(x)*p(x) [since F'(x)/F(x)=p(x)/q(x) → F'(x) = F(x)*p(x)/q(x) = G(x)*p(x)]

(9) Since p(x) and q(x) do not have any common divisors, it follows that G(x) is the greatest common divisor of F(x) and F'(x)

(10) Since we can always figure out the greatest common divisor of two equations (see Theorem 1, here for the greatest common divisor of polynomials), it follows that G(x) is obtainable based solely on F(x) and F'(x).

(11) Now since F(x)=G(x)*q(x), it follows that F(x)=0 divides into two equations:

G(x)=0

q(x)=0

(12) Since q(x) = (x - α)*(x - β)*(x - γ)*..., it is clear that it consists only of simple roots [from step#1]

(13) Since we can apply the same logic to G(x), it follows that we can always break down an equation of multiple real roots into a set of equations with only simple real roots.

QED

References

Monday, October 27, 2008

Jacques Charles Francois Sturm

Jacques Charles Francois Sturm was born on September 29, 1803 in Geneva, Switzerland. His father was a math teacher. When Sturm was 16, his father died and his family fell into a difficult financial situation.

At the Geneva Academy, Sturm's strong mathematical ability was recognized by his instructors. One of his teachers, Jean-Jacques Schaub, arranged financial support for young Sturm so he could attend school full time. At the Geneva Academy, Sturm met Daniel Colladon whose friendship and collaboration was an important part of his early work in mathematics.

When Sturm graduated from the Academy, he accepted a position as the tutor to Madame de Stael's youngest son in 1823. Madam de Stael had been a very successful and famous French writer who had died in 1817.

The family spent six months each year in Paris and Sturm was able to join them. Through the family, he was able to meet many of the intellectual luminaries of French society including Dominique Francois Jean Arago, Pierre-Simon Laplace, Simeon Denis Poisson, Jean Baptiste Joseph Fourier, Joseph Louis Gay-Lussac, and Andre Marie Ampere among others.

In 1824, Sturm and Colladon attempted to win a prize offered by the Paris Academy on the compressibility of water. The results were not as expected and Colladon severely injured his hand. They tried again in 1825. This time Sturm got a job tutoring Arago's son and was able to use Ampere's laboratory and received support and advice from Fourier. With all this new help, even if they did not win, they had made significant improvement from the previous year.

The next year, both Sturm and Colladon worked as assistants to Fourier. Additionally, they continued their experiments on the compressability of water and this time, they won the Grand Prix of the Academies de Sciences. The prize money was enough that they could stay in Paris and devote themselves to their research.

In 1829, Sturm published what would become one of his most famous papers: Mémoire sur la résolution des équations numériques. In it, he presented a major simplification of a method discovered by Cauchy to identify the number of real roots that an equation had over a specified interval. His method was largely based on methods from Fourier but the result was undeniably impressive. Her is Hermite's response:
Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever. His demonstration, which utilises only the most elementary considerations, is a rare example of simplicity and elegance.
Despite the well-received paper, Sturm had trouble finding work until the revolution of 1830. With the help of Arago, Sturm became professor of mathematics at the College Rollin. Three years later, he became a French citizen and three years after that he was admitted to the Academie des Sciences.

He would make significant contributions to differential equations relating to Poisson's theory of heat. Today, this work along with with the work done by Liouville form what is known as Sturm-Liouville Theory. Later in his career, he was professor at the Ecole Polytechnique. He made contributions to infinitesimal geometry, projective geometry, differential geometry, and geometric optics.

He died on December 18, 1855 in Paris.

References