Mathematicians have discovered a revolutionary method of sorting out higher-order polynomial equations. They hope that this represents a major turning point in the algebra landscape. This is a surprising breakthrough that has just been reported in the journal The American Mathematical Monthly on April 8. This finding hopes to make easier the age-old search for a master equation that can crack every polynomial equation.
Polynomials, one of humanity’s oldest and finest math inventions, are algebraic expressions that have their wealth of properties due to this nativity. Their origins reach all the way back to the first ancient civilizations such as Egypt and Babylon where early mathematicians struggled with their challenges. The roots of exponential numbers are the origins of radicals, and from them you can solve two-, three-, and four-degree polynomials! These strategies fail for polynomials of degree 4 and higher.
Higher-order polynomials, by which we mean those with variables raised to a power greater than four, have been particularly challenging. The solutions to these traditional methods can be messy, growing ever more complex with each increase in the degree of the polynomial.
In the video educator Norm Wildberger and his team have upended classical approaches in their groundbreaking alteration. They came up with an entirely different approach that avoids the use of radicals and irrational numbers altogether. This required some creative thinking, for which they derived their methodology from the Catalan numbers. This numeric sequence enumerates the triangulations of a given polygon. Mongolian-born mathematician Mingantu was the first to describe the Catalan numbers, around 1730. Independently, later, in 1751, Leonhard Euler discovered them.
Wildberger and his frequent collaborator Rubine recognized an opportunity. They noticed that the higher analogues of the Catalan numbers can be very useful in determining information in solving complicated polynomial equations. We can begin to understand the potential for new, previously unimaginable solutions in algebra with this innovative approach.
“This is a dramatic revision of a basic chapter in algebra.” – Norman Wildberger
Though their approach holds great potential, Wildberger warned that one should not hope for a magic bullet equation. He acknowledged the inherent complexity of higher-order polynomials, stating, “You would need an infinite amount of work and a hard drive larger than the universe.”
This new technique means so much more than just academics. Perhaps most importantly, it has the potential to shift how mathematicians approach polynomial equations in their postdocs and professorships. As the mathematical community evaluates this method’s potential, it stands as a testament to the ongoing evolution of algebra and its foundational concepts.
