Volumes of n-simplices with vertices on a polynomial space curve

TL;DR

This paper derives formulas for the volumes of n-simplices with vertices on polynomial space curves, extending the known area formula for triangles, using induction and symmetric polynomial identities.

Contribution

It introduces a generalized formula for the volumes of n-simplices on polynomial space curves, expanding geometric understanding.

Findings

Derived a volume formula for n-simplices on polynomial curves

Extended the area formula for triangles to higher dimensions

Utilized induction and symmetric polynomial identities in proofs

Abstract

In this paper, we give a formula for the area of the triangle formed by the vertices that live on a given polynomial, and we generalize this formula to the volumes of $n$-simplices with vertices on a polynomial space curve. To prove these results, we use induction arguments and a well known identity for complete symmetric polynomials.