Polynomial equations are fundamental to modern science, serving as a mathematical foundation for a wide array of applications including celestial mechanics, computer graphics, and market growth predictions. While most high school students can solve basic polynomial equations, the solutions to more complex, higher-order polynomials have remained a challenge even for experienced mathematicians. However, a groundbreaking breakthrough has emerged from the University of New South Wales, where mathematician Norman Wildberger and independent computer scientist Dean Rubine have developed the first general method for solving these intricate equations. Their findings were published on April 8 in the journal The American Mathematical Monthly.
A polynomial is an algebraic equation that consists of variables raised to a non-negative integer power, such as the equation x² + 5x + 6 = 0. The concept of polynomials dates back to ancient civilizations, including Egypt and Babylon, making it one of the oldest mathematical ideas. While mathematicians have long understood how to solve simple polynomials, those with degrees higher than four present significant difficulties.
The conventional methods for solving polynomials of degree two, three, and four typically involve the use of roots of exponential numbers known as radicals. Unfortunately, radicals often lead to irrational numbers, which are decimals that continue infinitely, such as the number pi. While it is possible for mathematicians to use radicals to approximate solutions for individual higher-order polynomials, finding a universal formula applicable to all of them has proven elusive. Wildberger explains, “Irrational numbers can never fully resolve. You would need an infinite amount of work and a hard drive larger than the universe.”
In their innovative approach, Wildberger and Rubine circumvented the use of radicals and irrational numbers altogether. Instead, they turned to polynomial extensions known as power series. Power series are theoretically infinite sequences of terms involving powers of x, commonly employed to tackle geometric problems. This technique falls under a specialized area of mathematics called combinatorics.
The mathematicians based their new method on the Catalan numbers, a sequence that represents the various ways to divide a polygon into triangles. This sequence was first identified by the Mongolian mathematician Mingantu around 1730 and was later rediscovered by the eminent mathematician Leonhard Euler in 1751. Wildberger and Rubine realized they could extend the Catalan numbers to address higher-order polynomial equations, a development they termed the Geode.
The Geode presents numerous potential applications for future research, particularly in the fields of computer science and graphics. Wildberger remarked, “This is a dramatic revision of a basic chapter in algebra,” highlighting the significance of their discovery. As mathematicians continue to explore the implications of this new method, the landscape of solving higher-order polynomial equations may be transformed, opening doors to new possibilities in both theoretical and applied mathematics.
