An equation deemed insoluble since 1832 finally resolved by Australian researchers

The history of mathematics is enriched by an extraordinary new chapter thanks to two Australian researchers who have just achieved the unthinkable. For almost two hundred years, the polynomial higher degree equations resisted the efforts of the most brilliant mathematical minds. This major discovery, published in June 2024 in The American Mathematical Monthlyrepresents much more than just resolutionresolution of equations, it could fundamentally redefine our understanding of mathematical structures and find applications in unsuspected scientific fields.

The bicentennial challenge of complex polynomial equations

Polynomial equations constitute a fundamental pillar of algebra, with their famous unknown X assigned to various exhibitors. Any student remembers having manipulated equations of the Ax² + Bx + C = 0 type during their schooling. On the other hand, the level of complexity increases considerably with the degree of the equation.

To understand the magnitude of the breakthrough, it is necessary to grasp the historical blocking to which the mathematiciansmathematicians. If the polynomial equations up to Degree 4 could be resolved by explicit formulas using radical expressions (square, cubic roots …), the degree and superior polynomials seemed to be condemned to remain in the dead end.

« The mathematical community had essentially accepted the impossibility of finding a general solution for these complex equationsexplains Norman Wildberger, professor at the University of New South Wales. Our unconventional approach made it possible to bypass this obstacle that many considered to be insurmountable ».

A revolutionary method inspired by geometry

The major innovation of Australian researchers lies in their approach radically different from the problem. Rather than beating on traditional methods, Wildberger and Rubine have studied an apparently unrelated mathematical territory: Catalan numbers.

These numbers, generally used to calculate the possible configurations of triangles in polygons, seemed a priori far from polynomial equations. However, the intuition of researchers has borne fruit.

« Our innovation consists in establishing a connection between the numbers of Catalan and the complex polynomial equationsspecifies Wildberger. We have identified that to resolve higher degree equations, it was necessary to develop complex equivalents with classic Catalan numbers ».

This transdisciplinary approach perfectly illustrates how major mathematical advances are often born from unexpected connections between different areas. The main characteristics of this method are:

• the use of geometric structures to solve algebraic problems;
• the extension of the numbers of Catalan to higher dimensions;
• a complete reformulation of the initial problem;
• The integration of advanced IT tools in resolution.

The unexpected discovery of “La Géode”

By continuing their research, the mathematicians made an even more surprising discovery. Beyond the resolution of the equations, they identified a new fundamental mathematical mathematical structure which they baptized “the geode”.

This structure seems to constitute the very foundation of Catalan numbers and could represent an even more significant conceptual breakthrough than the resolution of polynomial equations. There GeodeGeode appears as a new mathematical paradigm whose implications largely exceed the initial framework of research.

Dean Rubine, co -author of the study and computer specialist, underlines: ” Fundamental mathematical structures such as geode are extremely rare. When they emergingemergingthey generally offer new perspectives on problems apparently unrelated ».

Potential applications beyond pure mathematics

The implications of this breakthrough greatly exceed the theoretical framework of mathematics. Researchers have already identified several scientific fields likely to benefit directly from these new methods:

1. Molecular biologyMolecular biology : modeling of structures ofARNARN and prediction to fold proteinprotein ;
2. CryptographyCryptography : development of algorithms of encryptionencryption more robust;
3. Artificial intelligenceArtificial intelligence : Optimization of deep neural networks;
4. Quantum physics: resolution of complex equations describing the subatomic systems.

This discovery recalls that fundamental mathematical advances, even when they seem abstract, often end up transforming our understanding of the physical world and our ability to develop new technologies.

While researchers continue to visit the properties of geode and refine their method, the scientific community is impatiently awaiting future developments in this major mathematical breakthrough which could well mark the beginning of a new era in the history of mathematics.