Generalization of the Siebeck-Marden-Northshield Theorem to n-th Degree Polynomials

This paper generalizes the Siebeck-Marden-Northshield theorem to real-rooted polynomials of degree n by utilizing the geometric structure of a regular simplex. Previously, the real roots of cubic and quartic polynomials have been interpreted as the x-coordinate projections of the vertices of an equilateral triangle and a regular tetrahedron, respectively. In this study, we theoretically prove that when an n-th degree polynomial has n real roots, it is possible to construct a regular simplex in (n-1)-dimensional space whose vertices have these roots as their first coordinates. To establish this, we derive necessary and sufficient conditions based on properties of the regular simplex, including rotational invariance, symmetry, and inner product relations. Furthermore, we express the circumradius and the range of root distribution in terms of the polynomial coefficients. This geometric approach provides a novel framework for visualizing and structurally analyzing the distribution of real roots of higher-degree polynomials.