## Numerical curves and their applications to algebraic curves

### Volume 121 / 1996

#### Abstract

Hermite interpolation by bivariate algebraic polynomials and its applications to some problems of the theory of algebraic curves, such as the existence of algebraic curves with given singularities, is considered. The scheme $N={n_1,..., n_s;n}$, i.e., the sequence of multiplicities of nodes associated with the degree of interpolating polynomials, is considered. We continue the investigation of canonical decomposition of schemes and define so called maximal schemes. Some numerical results concerning the factorization of schemes are established. This leads to determination of irreducibility or to finding the (exact) number of components of algebraic curves as well as to the characterization of all singular points of a wide family of algebraic curves. Also, the Hilbert function of schemes is discussed. At the end, the problem of regularity of schemes depending on the number of interpolation conditions is considered.