By Alexander V. Mikhalev, A.B. Levin, E.V. Pankratiev, M.V. Kondratieva

ISBN-10: 9048151414

ISBN-13: 9789048151417

ISBN-10: 9401712573

ISBN-13: 9789401712576

The function of Hilbert polynomials in commutative and homological algebra in addition to in algebraic geometry and combinatorics is celebrated. an analogous function in differential algebra is performed through the differential size polynomials. The inspiration of differential measurement polynomial was once brought by way of E. Kolchin in 1964 [KoI64]' however the difficulties and ideas that had ended in this idea (and which are mirrored during this publication) have primarily extra lengthy historical past. truly, it is easy to say that the differential measurement polynomial describes in detailed phrases the liberty measure of a dynamic approach in addition to the variety of arbitrary constants within the normal answer of a procedure of algebraic differential equations. the 1st makes an attempt of such description have been made on the finish of nineteenth century by means of Jacobi [Ja890] who anticipated the variety of algebraically self sufficient constants within the common resolution of a method of linear traditional differential equations. afterward, Jacobi's effects have been prolonged to a few circumstances of nonlinear structures, yet typically case the matter of such estimation (that is called the matter of Jacobi's certain) continues to be open. There are a few generalization of the matter of Jacobi's guaranteed to the partial differential equations, however the leads to this sector are only showing. at first of the twentieth century algebraic tools within the concept of differen tial equations have been actively constructed through F. Riquier [RiqlO] and M.

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Let f : R -+ R' be a ring epimorphism and J = Ker f. If E is a conservative system of ideals of R, then E' = {1 EEl 1 :2 J} is also a conservative system of ideals of R, and the mapping 1 -+ f(I) (1 E E') is an isomorphism of E' onto a conservative system of ideals of R' which is denoted by f(E). If f is a natural ring epimorphism R -+ Rj J (where J is an ideal of R), then the conservative system f(E) is denoted by Ej J and the isomorphism of conservative systems E' = {1 E E I 1 :2 J} and Ej J is called canonical.

55 to the finitely generated A-module M = A[X1, ... , Xk] (it is easy to see that xn M S; 1M). 65. LEMMA. Let A be an integrally closed domain with a quotient field K and let f(X) = xm + aOX m- 1 + ... + am, g(X) = xn + bl X n - 1 + ... + bn be polynomials in one indeterminate X over K. Then f(X)g(X) E A[X] if and only if f(X) E A[X] and g(X) E A[X]. PROOF. 9 E A[X] implies f E A[X] and 9 E A[X]. Let L be a field extension of K in which f and 9 can be factorized into a product of linear factors: f(X) = 1(X - Oi), g(X) = n;=l (X - (3j).

22. EXERCISE. Let us consider a standardly graded polynomial ring R = K[XI' ... ' xnl over a field K as a graded module over R. Show that the Hilbert function

### Differential and Difference Dimension Polynomials by Alexander V. Mikhalev, A.B. Levin, E.V. Pankratiev, M.V. Kondratieva

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