Chebyshev polynomials are useful in function approximation as the root finding problem is better-conditioned in the basis of Chebyshev polynomials than in the familiar monomial basis. In this talk, I will introduce multivariate generalizations of Chebyshev polynomials and use them to define Chebyshev varieties parametrized by Chebyshev polynomials analogous to toric varieties parametrized by monomials. I will also discuss the geometry of Chebyshev varieties and list some applications. This talk is based on the paper Chebyshev varieties by Z. Bel-Afia, C. Meroni and S. Telen ( arXiv:2401.12140 ).

Following this presentation, there will be a more general discussion of arithmetic toric varieties.