Order and degree bounds for the differential Nullstellensatz

LISI D'ALFONSO; GABRIELA JERONIMO; PABLO SOLERNÓ

Seúl

Congreso; International Congress of Mathematicians; 2014

International Mathematical Union

A differential version of Hilbert´s Nullstellensatz was introduced by Ritt in 1932: if f1,..., fs, g are multivariate differential polynomials with coefficients in an ordinary differential field K such that every zero of the system f1,..., fs in any differential extension of K is a zero of g, then some power of g is a linear combination of the fi´s and a certain number of their derivatives, with polynomials as coefficients. This result was later extended to arbitrary differential fields. The first known bound for orders of derivatives in the differential Nullstellensatz for both partial and ordinary differential fields was given in 2008 by Golubitsky et al, but this bound is not an elementary recursive function of the number of variables, the number of polynomials, their orders and degrees. We present new order and degree bounds for the differential Nullstellensatz in the case of ordinary systems of DAE equations over a field of constants of characteristic 0. Our main result is a doubly exponential upper bound for the number of successive derivatives involved. Combining this upper bound with effective versions of the classical algebraic Hilbert´s Nullstellensatz, we also obtain a bound for the power of g in the differential ideal and for the degrees of polynomial coefficients in a linear combination of f1,...fs and their derivatives representing this power of g.