Marina V. Kondratieva

A bound for orders in differential Nullstellensatz

Abstract We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed in [A. Seidenberg, An elimination theory for differential algebra, Univ. of California Publ. in Math. III (2) (1956) 31–66] but no complete solution was given. Our result is a complement to the corresponding result in algebraic geometry, which gives a bound on degrees of polynomial coefficients in effective Nullstellensatz [G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1) (1926) 736–788; E.W. Mayr, A.W. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math. 46 (3) (1982) 305–329; W.D. Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. 126 (3) (1987) 577–591; J. Kollar, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (4) (1988) 963–975; L. Caniglia, A. Galligo, J. Heintz, Some new effectivity bounds in computational geometry, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Rome, 1988, in: Lecture Notes in Comput. Sci., vol. 357, Springer, Berlin, 1989, pp. 131–151; N. Fitchas, A. Galligo, Nullstellensatz effectif et conjecture de Serre (theoreme de Quillen–Suslin) pour le calcul formel, Math. Nachr. 149 (1990) 231–253; T. Krick, L.M. Pardo, M. Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 109 (3) (2001) 521–598; Z. Jelonek, On the effective Nullstellensatz, Invent. Math. 162 (1) (2005) 1–17; T. Dube, A combinatorial proof of the effective Nullstellensatz, J. Symbolic Comput. 15 (3) (1993) 277–296]. This paper is dedicated to the memory of Eugeny Pankratiev, who was the advisor of the first three authors at Moscow State University.

On the generalized Ritt problem as a computational problem

The Ritt problem asks if there is an algorithm that decides whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing whether a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of this equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking.

Computation of Dimension Polynomials

The first partial implementation by the authors of the described algorithms was made in 1980 [MP80]. The programs were written in the algorithmic language REFAL, run on the computer BESM-6 and were destinated to compute characteristic sets of differential ideals in the ring of differential polynomials.

Algebraic transformation of differential characteristic decompositions from one ranking to another

Canonical characteristic sets of characterizable differential ideals are studied. For the ordinary case, an estimate of orders of elements from the canonical characteristic set is proved. It is also shown how one can verify the equality of characterizable ideals without using canonical characteristic sets. DOI: 10.3103/S0027132208020095We propose an algorithm for transforming a characteristic decomposition of a radical differential ideal from one ranking into another. The algorithm is based on a new bound: we show that, in the ordinary case, for any ranking, the order of each element of the canonical characteristic set of a characterizable differential ideal is bounded by the order of the ideal. Applying this bound, the algorithm determines the number of times one needs to differentiate the given differential polynomials, so that a characteristic decomposition w.r.t. the target ranking could be computed by a purely algebraic algorithm (that is, without further differentiations). We also propose a factorization-free algorithm for computing the canonical characteristic set of a characterizable differential ideal represented as a radical ideal by a set of generators. This algorithm is not restricted to the ordinary case and is applicable for an arbitrary ranking.

A bound for the Rosenfeld–Gröbner algorithm

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. We also give an algorithm for converting a characteristic decomposition of a radical differential ideal from one ranking into another. This algorithm performs all differentiations in the beginning and then uses a purely algebraic decomposition algorithm.

Membership problem for differential ideals generated by a composition of polynomials

The question of whether a polynomial belongs to a finitely generated differential ideal remains open. This problem is solved only in some particular cases. In the paper, we propose a method, which reduces the test of membership for fractional ideals generated by a composition of differential polynomials to another, simpler, membership problem.

An upper bound for minimizing coefficients of dimension Kolchin polynomial

In this paper, an upper bound for minimizing coefficients of the dimension Kolchin polynomial for a subset E ⊂ ℕ0m, which depends on the maximal order of elements in E, is obtained. The minimizing coefficients are always positive; some of them are invariant and play an important role in differential algebra. As an example of application of the result obtained, an estimate for a typical differential dimension of a system of partial differential equations is obtained in the case where the orders and degrees of the equations are bounded.

Characteristic sets for ordinary differential equations

In this paper, the problem of the construction of a characteristic set in the sense of Kolchin for a radical differential ideal is considered. Algorithms for constructing such sets in the ordinary case for arbitrary radical differential ideals, which are based on the estimate of the orders of their elements, are presented. These algorithms are applicable in the case of an orderly ranking on the set of the derivatives. Advantages of the regular and characteristic decompositions of radical differential ideals are discussed.

Computation of the Differential Dimension Polynomial When Changing Generators in the Dirac Equations

..., of the degree less than or equal to s and trdeg is the transcendence degree of the field extension. In other words, all the derivatives of the elements φ 1 , ..., φ k of the degrees 1, ..., s are adjoined to the initial field F , and the cardinality is computed for the maximal algebraically independent over F , set of elements of the field obtained. For sufficiently large s , this dependence is polynomial, or, more precisely, the differential dimension polynomial is the Hilbert polynomial of the filtered module of differentials of the extension G over F . This polynomial contains some ∆ -invariants of the field G over F (in particular, the degree and the leading coefficient), but the polynomial itself may change if another system of generators F 〈ψ 1 , ..., ψ l 〉 of G is chosen. If the degree of ( s ) is less than m and the

Basic Notions of Differential and Difference Algebra

Let R be a ring and let ∆ be a set of operators acting on R. In this case R is said to be a ∆-ring and ∆ is called its basic set of operators. In the following sections the operators in ∆ will be either derivation operators or endomorphisms but now we do not impose any restrictions on ∆.