Chebyshev curves and Hodge theory
Nice University, France.
Let \(T_d(x)=\cos (d \arccos (x))\) be the classical Chebyshev polynomial of degree \(d\). We consider the projective complex plane curve \(C_d\) with affine equation \(T_d(x)+T_d(y)=0.\) We discuss the irreducible components of \(C_d\) and the Hilbert-Poincar\'e polynomial of the associated Jacobian ideal. Using Hodge theory in the form of Hodge-Deligne polynomials, we show that our results on the Chebyshev curves \(C_d\) turn out to be quite general.