1. Given any set of polynomials S, its zero set is the set V (S) = {x ? R d : f (x) = 0 for all f ? S}. The zero set of the family R coincides with the zero set of the ideal I = (R), that is, V (R) = V (I). For example, the set M in (5.2) is the zero set of the ideal (Q). Given a set V ? R d , the ideal generated by V , denoted by I (V ), is the set of all polynomials that vanish on V . It follows from the definition that S ? I (V (S)) for any set of polynomials S. A basic problem in algebraic geometry is to establish when an ideal;An ideal I of Pol(R d ) is a subset of Pol(R d ) closed under addition such that f ? I and g ? Pol(R d ) implies f g ? I. Given a finite family R = {r 1 , . . . , r m } of polynomials, the ideal generated by R, denoted by (R) or (r 1 , . . . , r m ), is the ideal consisting of all polynomials of the form f 1 r 1 + � � � + f m r m , with f i ? Pol(R d )
2. Linear Credit Risk Models
3. The Jacobi stochastic volatility model