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Last Updated: Thursday, 27 November, 2003, 15:22 GMT
Historic maths problem 'cracked'
By Dr David Whitehouse
BBC News Online science editor

Elin Oxenhielm, Stockholm University
Elin Oxenhielm chalks up a solution
A 22-year-old student at Stockholm University, Elin Oxenhielm, may have cracked part of one of mathematics' greatest unsolved problems.

Called Hilbert's problem 16, it has confounded workers for over a century.

But in just a few hours of inspiration, Oxenhielm saw the light. Her solution is to be published in a maths journal.

Her research into so-called planar polynomial vector fields may have practical applications for computer simulations in science and economics.

Passion for maths

"I solved it before I knew its significance," Elin Oxenhielm told BBC News Online.

"It took a few months of thinking about it at first, but then the solution came remarkably quickly," she says.

Her breakthrough comes a century after the problem was posed by Prussian mathematician David Hilbert. In 1900 he gave a lecture in Paris where he laid down the 23 greatest problems for maths in the 20th Century.

They were a varied selection that had confounded the greatest mathematical minds of the age.

Couched in language that only mathematicians appreciate, they included such questions as: can the continuum of numbers be regarded as a well ordered set, and can space be constructed by congruent polyhedra?

Over a century later only three of Hilbert's problems remain unconquered, numbers six, eight and 16.

Number eight is the famous Riemann hypothesis to do with prime numbers, regarded by many as the most difficult problem in maths today.

Recently problems eight and 16 on Hilbert's list have been placed on a list of the 18 biggest challenges for 21st Century mathematicians.

But problem 16 may now have a partial answer.

A few hours' work

Mathematicians describe it as a question of the "topology of algebraic curves and shapes." Non-technically it deals with the way solutions to equations are arrived at.

Elin Oxenhielm's solution is of a special version of the second part of the problem, called the "boundary cycles for polynomial differential equations".

"It only took me a few hours to solve the problem once I expressed it in the correct way," she said.

"It is difficult to describe for non-mathematicians but the way I solved the problem may have practical applications."

It may improve the way scientists use computers to simulate such diverse phenomena as global warming and economies.

Her work is to be published in the mathematical journal Nonlinear Analysis.

Oxenhielm believes her method can be used to unlock the mystery of the entire 16th problem, and plans to write a popular book about her work.

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