Computing generators of the ideal of a smooth affine algebraic variety

CRISTINA BLANCO; GABRIELA JERONIMO; PABLO SOLERNÓ

JOURNAL OF SYMBOLIC COMPUTATION

Elsevier

Año: 2004 vol. 38 p. 843 - 872

0747-7171

Let K be an algebraially closed field, V in K^n be a smooth equidimensional algebraic variety and I(V) in K[x_1,...,x_n] be the ideal of all polynomials vanishing on V. We show that there exists a system of generators f_1,..., f_m of I(V) such that m =<  (n - dim V)(1 + dim V) and deg(f_i) =< deg V for i = 1,..., m. If char(K) = 0 we present a probabilistic algorithm which computes the generators f_1,..., f_m from a set-theoretical description of V. If V is given as the common zero locus of s polynomials of degrees bounded by d encoded by straight-line programs of lenght L, the algorithm obtains the generators of I(V) with error probability bounded by \varepsilon within complexity s (nd^n)^{O(1)} log^2([1/\varepsilon]) L.