Algorithms yield upper bounds in differential algebra

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields). We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

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1. Introduction

Finding uniform bounds for problems and quantities (e.g., consistency testing or counting of solutions) is one of the central questions in differential algebra. In [27], it was demonstrated that, in commutative algebra, one can show the existence of such bounds as a consequence of theorems about nonstandard extensions of standard algebraic objects. This approach was successfully applied in the differential algebra context in [11] and [8, Section 6] for establishing, for example, the existence of a uniform bound in the differential Nullstellensatz. Furthermore, in [26], the authors used methods of proof theory to extract explicit bounds based on nonstandard existence proofs.

The present paper can be viewed as an alternative approach, in which we derive the existence of a computable uniform bound for an object from the existence of an algorithm for computing the object. More precisely, let be a complete decidable theory. The most relevant examples for us would be the theory of differentially closed fields in zero characteristic with commuting derivations and the theory of existentially closed difference fields, others include algebraically closed and real closed fields. Consider an algorithm performing computations in a model of that is restricted to using only definable functions when working with elements of the model (for formal definition, we refer to Section 4.1) and required to terminate for every input.

We show that there is a computable upper bound for the size of the output of in terms of the input size of . We apply this to the Rosenfeld-Gröbner algorithm [3] that decomposes a solution set of a system of polynomial PDEs into components and is such an algorithm. This allows us to show that there is a uniform upper bound for the number of components of any differential-algebraic variety defined by a system of polynomial PDEs. We also show how this bound for the number of components leads to a uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

A bound for the number of components of varieties defined by polynomial ODEs appeared in [18], as did a bound for the elimination problem for polynomial ODEs with delays. These bounds are based on the application of the Rosenfeld-Gröbner algorithm, which, if applied in this situation to ODEs, outputs equations whose order does not exceed the order of the input. This allowed to restrict to a finitely generated subring of the ring of differential polynomials and use tools from algebraic geometry. It is non-trivial to generalize this to polynomial PDEs because the orders in the output of the Rosenfeld-Gröbner can be greater than the orders of the input. Another key ingredient in the ODE case to obtain the bound in [18] was an analysis of differential dimension polynomials. A significant difference of our present PDE context with the ordinary case that these polynomials behave less predictably under projections of varieties (compare [18, Lemma 6.16] and Lemma 6.6). To overcome this difficulty, we use again our bound for the Rosenfeld-Gröbner algorithm.

We believe that our method can also be applied to obtain bounds for other algorithms in differential algebra such as [1, Algorithm 3.6] and for algorithms from other theories, e.g. [7, Algorithm 3] for systems of difference equations. Since the reducibility of a polynomial can be expressed as a first-order existential formula, it seems plausible that the same methods could be applied to other algorithms dealing with difference [5] and differential-difference [6] equations that use factorization because the corresponding theories satisfy the requirements of our approach [14, 16, 23]. However, we leave these for future research.

The paper is organized as follows. Section 2 contains definitions and notation used in Section 3 to state the main results. Bounds for an algorithm working with a model of a theory are established in Section 4. These results are applied to differential algebra in Section 5. Further applications to delay PDEs are given in Section 6.

2. Basic notions and notaiton

Definition 2.1 (Differential-difference rings).
  • A --ring is a commutative ring endowed with a finite set of commuting derivations of and an endomorphism of such that, for all , .

  • When is additionally a field, it is called a --field.

  • If is an automorphism of , is called a --ring.

  • If , is called a -ring or differential ring.

  • For a commutative ring , denotes the ideal generated by in .

  • For , let .

  • For , we let . For a non-negative integer , we denote .

  • For a -ring , the differential ideal generated by in is denoted by ; for a non-negative integer , we introduce the following ideal of :

Definition 2.2 (Differential polynomials).

Let be a -ring. The differential polynomial ring over in is defined as

The structure of a -ring is defined by for every .

Definition 2.3 (Differential-difference polynomials).

Let be a --ring. The differential-difference polynomial ring over in is defined as

The structure of - ring is defined by and for every and .

A --polynomial is an element of . Given , let denote the polynomial ring

The notions from logic that we use are described in detail in [19]. In particular, we will use the notions of a first-order language [19, Definition 1.1.1], structure [19, Definition 1.1.2], formula [19, Definition 1.1.5], theory [19, Section 1.2, page 14], model [19, Section 1.2, page 14], compactness [19, Section 2.1], complete theory [19, Definition 2.2.1], decidable theory [19, Definition 2.2.7], quantifier elimination [19, Definition 3.1.1], and -saturation [19, Definition 4.3.1].

3. Main results

For clarity, we gather our main results in one section.

Theorem 3.1 (Upper bound for irreducible components for PDEs).

There exists a computable function such that, for every differential field of zero characteristic with a set of commuting derivations and finite with , the number of components in the variety defined by does not exceed .

Additional details and proof are given in Theorem 5.13.

Theorem 3.2 (Upper bound for elimination in delay PDEs).

For all non-negative integers , and , there exists a computable such that, for all:

  • non-negative integers and ,

  • a --field with and ,

  • sets of --polynomials , where , , and ,

we have

Corollary 3.3 (Effective Nullstellensatz for delay PDEs).

For all non-negative integers , and , there exists a computable such that, for all:

  • --fields with and ,

  • sets of --polynomials , where , and ,

the following statements are equivalent:

  1. There exists a - field extending such that has a sequence solution in .

  2. .

  3. There exists a field extension of such that the polynomial system in the finitely many unknowns has a solution in .

The two preceding theorems are proved using our main technical result about algorithms performing computations in complete decidable theories. Stating it precisely requires defining admissible algorithms carefully, so we postpone it until Section 4 and give here a simplified and informal version of the statement.

Theorem 3.4 (Algorithm yields a bound, stated precisely as Theorem 4.5).

There exists a computable function with input

  • a complete decidable theory ;

  • an algorithm performing computations in a model of restricted to using only definable functions when working with elements of the model;

  • positive integer

that computes a number such that for every model of and every the size of the output of with input does not exceed .

For the application of this to the Rosenfeld-Gröbner algorithm, see Theorem 5.10.

4. Bounds for the output size of algorithms over complete theories

In this section, we will use the formalism of oracle Turing machines 

[24, § 14.3]. Roughly speaking, an oracle Turing machine is a Turing machine with an extra tape for performing queries to an external oracle. An oracle is not considered to be a part of the machine.

4.1. Setup

To consider an algorithm dealing with elements of a (not necessarily computable) model of a theory , we will “encapsulate” the elements of the model given to the algorithm into an oracle that allows to perform only first-order operations with them as defined below. Alternatively, one could adapt other approaches used to formalize computations in real numbers [2, Section 3] or in arbitrary structures (see [9, §1] and [4, §2.2]).

Definition 4.1 (-oracle).

Let be a language and be a theory in . For elements of a model of , any oracle that supports the following queries: given a formula , the oracle returns the value in (can be true or false), will be denoted by and called an evaluation oracle.

Definition 4.2 (Total algorithm over ).

An oracle Turing machine will be called a total algorithm over if, for all positive integers , every model of and every , the machine with every input and oracle is guaranteed to terminate.

4.2. Auxiliary bound and result

Lemma 4.3.

There is an algorithm that takes as input:

  • language ;

  • a complete decidable theory given by a Turing machine checking correctness of sentences in the theory;

  • a total algorithm over ;

  • positive integers and ;

  • a string in the input alphabet of ;

and computes

  • a first-order formula in in variables and

  • a number

such that, for any model of and tuple , the following are equivalent:

  1. the sentence is true in ;

  2. algorithm with input and oracle terminates after performing at most queries to the oracle

and if these statements are true, then the number of steps performed by with input and oracle does not exceed .

Proof.

We describe an algorithm for computing and . Fix some , and .

We will describe an algorithm that, for a given positive integer , computes first-order formulas and in in the variables and a positive integer such that, for every model of and every

  • is true in iff algorithm with input and oracle will perform at least queries;

  • if is true in , then the result of the -th query will be ;

  • if algorithm with input and oracle performs at most queries, then the number of steps performed does not exceed .

Fix some and assume that the algorithm have computed , , and . Assume that with input has performed queries. Then whether or not an -th query will be performed is determined by the results of the first queries. Fix some . It will represent possible results of the first queries. Consider the following formula in :

where we assume . The algorithm uses the algorithm for checking correctness of sentences in to check whether the sentence is false in . If it is, then there is no oracle of the form such that will perform at least queries on it with the results being .

In the case of is true in , the algorithm will run with input and an oracle that works as follows. For the first queries, will return . For all subsequent queries, it always returns True. The algorithm will stop the execution of if makes an -th query to the oracle, and denote the formula in the query by .

Since is true in , gives the same responses to the first queries as some oracle of the form . Since must terminate in finite time for every such oracle, one of the following must happen:

  1. will perform an -th query.

  2. will terminate after performing only queries.

In the former case, as described above, the algorithm will define a formula to be the -th query. In the latter case, the algorithm will define to be the number of steps performed by . Then the algorithm computes

where we assume . If the set is empty, the algorithm sets and . Finally, the algorithm returns and . ∎

Lemma 4.4.

Let be a theory and an -saturated model. Let be a sequence of definable sets in such that . Then there exists such that .

Proof.

Assume the contrary, that is, that for every . We will show that .

We show that a collection of formulas is finitely satisfiable. Indeed, let be a finite set and . Then . Due to compactness, the countable collection is satisfiable in some elementary extension of . Since is -saturated, this collection is satisfiable in . Therefore, . ∎

4.3. Main result

Theorem 4.5.

There exists a computable function with input

  • a complete decidable theory (given by an algorithm for checking correctness of sentences);

  • a total algorithm over ;

  • positive integers and

that computes a number such that for every model of , every , and every string in the alphabet of of size at most , the number of steps performed by with input and oracle does not exceed .

Remark 4.6.

Let the intermediate result at step for a total algorithm with given input and oracle be the content of all the cells of the tape that have been read by the Turing machine. Since a Turing machine can read at most one cell at each step, the number of these cells cannot exceed . Therefore, the intermediate result at step can be encoded using bits, where is the cardinality of the alphabet of . In particular, if a binary alphabet is used, the bitsize of the intermediate result never exceeds the total number of steps in the algorithm.

Proof.

We will describe an algorithm for computing . We fix , , , and . We will consider of length at most and describe how to compute a bound for the number of steps given that the input is . Taking the maximum over all of length at most (there are finitely many of them), we obtain .

The algorithm will compute for using the algorithm from Lemma 4.3. For each , the algorithm will check whether the formula is equivalent to in using the decidability of .

If this is true, the algorithm stops and returns (see Lemma 4.3). It remains to show that the described procedure terminates in finitely many steps. Let be an -saturated model of (it exists, for example, due to [19, Theorem 4.3.12]). For every , we introduce a definable set

Notice that if and only if in . Then the definition of ’s implies that . Assume that is not empty and choose an element in it. Then will not terminate in finitely many steps with input and oracle . Thus, . Lemma 4.4 implies that there exists such that . Then our algorithm will terminate after checking whether  is equivalent to True. ∎

5. Applications to differential algebra

In this section, we will apply the results of Section 4 to the theory of differentially closed fields with several commuting derivations.

5.1. Preparation

Notation 5.1.

Let be a positive integer.

  • The language of partial differential rings with commuting derivation is denoted by . We add a separate functional symbol for subtraction for convenience.

  • The theory of partial differentially closed fields with commuting derivations of characteristic zero is denoted by . Recall that is complete [21, Corollary 3.1.9 ] and, with this, is decidable by [19, Lemma 2.2.8] and [21, Lemma 3.1.2 and page 890].

Notation 5.2.

Let be positive integers and a differential field with a set of commuting derivations .

  • denotes the space of all differential polynomials over in variables of order at most and degree at most .

  • The dimension of (which does not depend on ) will be denoted by .

Notation 5.3.

Let , and be positive integers.

  • Let denote the ring of differential polynomials in differential variables with respect to derivations with the coefficients being terms in the language in (that is, elements of .

    This is a computable differential ring with commuting derivations. In what follows, we will assume that the algorithms use dense representation to store these polynomials (that is, store all the coefficients up to certain order and certain degree).

  • Let be a differential field with derivations and . Then, for , we define to be the result of evaluating the coefficients of at .

Definition 5.4.

A differential ranking for is a total order on satisfying, for all , :

  • for all , and

  • for all , if , then .

Notation 5.5.

For a -field and and differential ranking ,

  • is the element of of the highest rank appearing in .

  • The leading coefficient of considered as a polynomial in is denoted by and called the initial of .

  • The separant of is .

  • The rank of is . The ranks are compared first with respect to , and in the case of equality with respect to .

  • For , the set of initials and separants of is denoted by .

Remark 5.6 (Defining a ranking).

In general, there are uncountable many differential rankings already for and . However, [25, Theorem 29] implies that any differential ranking can be defined by real numbers together with integers not exceeding and one permutation on elements. We define a function taking as input a tuple of real numbers and a binary string (of length at most ) encoding the integers and the permutation and returning the corresponding binary predicate on the derivatives as in [25, Definition 28]. The relevant properties of this encoding for us will be that, for fixed :

  1. the statement that defines a ranking is a first-order formula in in the language of ordered fields;

  2. for every two derivatives and , the fact that with respect to is also a first-order formula in in the language of ordered fields.

Definition 5.7 (Characteristic sets).
  • For , is said to be reduced w.r.t. if no proper derivative of appears in and .

  • A subset is called autoreduced if, for all , is reduced w.r.t. every element of . One can show that every autoreduced set is finite [13, Section I.9].

  • Let and be autoreduced sets ordered by their ranks (see Notation 5.5). We say that if

    • and , , or

    • there exists such that and, for all , , .

  • An autoreduced subset of the smallest rank of a differential ideal is called a characteristic set of . One can show that every non-zero differential ideal in has a characteristic set.

  • A radical differential ideal of is said to be characterizable if has a characteristic set such that .

The Rosenfeld-Gröbner algorithm [3, Theorem 9] takes as input a finite set of differential polynomials and a differential ranking and outputs autoreduced sets such that

(1)

and that, for each , , is a characteristic set of . The representation (1

) can be used, for example, for membership testing, estimating the number of irreducible components (used in Theorem 

5.13) or the Kolchin polynomial (used in Section 6) of a differential-algebraic variety.

With the next Proposition 5.9 we express how we will call the Rosenfeld-Gröbner algorithm. This algorithm depends on the choice of a differential ranking. The reader may wish to make one such choice once and for all, thereby ignoring the potential ambiguity. However, since such a choice may affect the size of the output and the efficiency of any given implementation of the algorithm, one may prefer to allow for these other orderings.

We will express this dependence by seeing the algorithm as a total algorithm relative to the two-sorted theory which is a disjoint union of and the complete decidable theory with quantifier elimination of real closed fields  [19, Theorem 3.3.15 and Corollary 3.3.16]. Then we will use the characterization of differential rankings via real numbers from Remark 5.6.

Lemma 5.8.

Theory is decidable and complete.

Proof.

In order to prove the completeness and decidability, we will prove that there is an algorithm for quantifier elimination in based on the existence of such algorithms for (follows from decidability, see Notation 5.1, and quantifier elimination [21, Theorem 3.1.7]) and . It is sufficient to perform quantifier elimination for a formula of the form

where is one of the sorts (corresponding to or ) and are literals. (See [19, Lemma 3.1.5].) By reordering if necessary, we will further assume that there exists such that are in the signature of the sort and are in the signature of the other sort. Then

and, for , the algorithm for the corresponding sort can compute an equivalent quantifier-free formula.

The resulting theory is decidable because the correctness of each sentence can be checked by performing quantifier elimination after which the formula will become just true/false. ∎

Proposition 5.9.

There is a computable function that, for a given positive integer , computes a total algorithm over such that, for every differential field with derivations and and any , the input-output specification of with oracle is the following:

Input:

finite subsets and of and a binary string ;

Output:

if (see Remark 5.6) defines a differential ranking, return a list of tuples from such that

is the output of the Rosenfeld-Gröbner algorithm [3, Theorem 9] with input with respect to the ranking . Otherwise, return .

Proof.

[3, Theorem 9] states that the only operations performed by the Rosenfeld-Gröbner algorithm with the elements of the ground differential field are arithmetic operations, differentiation, and zero testing. Algorithm is constructed to work exactly in the same way as the Rosenfeld-Gröbner algorithm with the only difference that the elements of the ground differential field will be represented as , where . The arithmetic operations and differentiations can be performed with , zero testing can be performed using the -component of the oracle, and the queries to the ranking can be performed using the -component of the oracle, so will be able to perform the same computations as the Rosenfeld-Gröbner algorithm.

Due to [3, Theorem 5], the Rosenfeld-Gröbner algorithm is guaranteed to terminate on every input. Hence, the same is true for . ∎

5.2. Bounds

Theorem 5.10 (Upper bound for Rosenfeld-Gröbner algorithm).

There exists a computable function such that, for every differential field with derivations and subsets with , and every differential ranking, the Rosenfeld-Gröbner algorithm [3, Theorem 9] on and will produce at most components with all the orders and degrees of the differential polynomials occurring in the algorithm not exceeding .

Proof.

We fix integers , , and and compute the total algorithm over from Proposition 5.9. Let be the set of all the coefficients of and . Then . The sets and can be presented as evaluations of subsets at such that the orders and degrees of in do not exceed and every coefficient is a single variable . Let the ranking be defined as (see Remark 5.6), where is a tuple of real numbers and is a binary string of length at most . Then the the tuple can be encoded as a binary string of the length bounded by a computable function .

We run with the input and oracle . Theorem 4.5 implies that the number of steps and, consequently, the bitsize of of all the intermediate results (see Remark 4.6) will not exceed .

Since each component takes at least one bit, a polynomial of degree or order has at least coefficients (due to the dense representation of the polynomials, see Notation 5.2) requiring at least one bit each, the number of components, the degrees and orders do not exceed the bitsize of the intermediate results. Therefore, we can set . ∎

Corollary 5.11.

There exists a computable function such that, for every computable differential field with derivations and subsets with , and every differential ranking, the ideal can be written as an intersection of at most characterizable differential ideals defined by their characteristic sets with respect to the ranking of order and degree not exceeding .

Proof.

Theorem 5.10 implies that there exists a representation

where is the product of the initials and separants of , and is the characteristic presentation [3, Definition 8] of for every . As noted in [3, p. 108] a characteristic set of can be obtained from by performing reductions until it will become autoreduced. Since differential reduction is a part of the Rosenfeld-Gröbner algorithm, it can also be performed by a total algorithm over . Therefore, as in the proof of Theorem 5.10, Lemma 4.5 implies that has a characteristic set with degrees and order bounded by a computable function of the degrees and orders of . The latter are bounded by a computable function due to Theorem 5.10. Composing these two bounds, we obtain a desired function . ∎

Lemma 5.12.

There exists a computable function such that for every partial differential field with derivations, every ranking, and every characterizable differential ideal defined by a characteristic set with respect to this ranking, we have

  1. the number of prime components of does not exceed ;

  2. every prime component of has a characteristic set with respect to the ranking with orders and degrees bounded by .

Proof.

Let be the product of the initials and separants of . [3, Theorem 4] implies that the number of prime components of is equal to the number of prime components of the algebraic ideal , where is the ring of differential polynomials of order at most . Since the degrees of elements of are bounded by , the Bézout inequality implies that there is a computable bound for the degree of the radical ideal (defined, e.g., as the degree of the corresponding affine variety [12, page 246]) in terms of and , so this gives a bound for the number of components.

Let be the prime components of . For every , is a prime algebraic ideal, and its zero set can be defined by equations of degree at most due to [12, Proposition 3]. Therefore, for each , we can choose a polynomial in of degree at most . Their product has degree at most . Observe that

Thus, applying Corollary 5.11 to a pair and using that , we show that has a characteristic set with orders and degrees bounded by . ∎

Theorem 5.13 (Upper bound for the components of a differential variety and their number).

There exists a computable function such that, for all non-negative integers , and and a partial differential field with derivations and finite set :

  1. the number of components in the variety defined by does not exceed ;

  2. for every differential ranking and every component of the variety , has a characteristic set with respect to the ranking with orders and degrees bounded by .

Proof.

Consider any differential ranking. By replacing with the basis of its linear span, we will further assume that (see Notation 5.2). Corollary 5.11 implies that can be represented as an intersection of at most characterizable ideals with characteristic sets of order and degree at most , where

Lemma 5.12 applied to each of implies that the number of components of the variety defined by does not exceed , and each of them has a characteristic set with orders and degrees not exceeding . ∎

Remark 5.14.

It was shown in [11, Theorem 6.1] that there exists a (not necessarily computable) bound for the degrees and orders a characteristic set of a prime differential ideal. The second part of Theorem 5.13 implies that there is a computable bound.

6. Application to delay PDEs

In this section, we will show how Theorem 5.13 applies to the problem of elimination of unknowns in delay PDEs.

The proof of the main result of this section, Theorem 6.23 (Effective elimination theorem for delay PDEs) inherited from [18] had only two missing ingredients closely related to each other: the bound on the number of components of the variety defined by a system of differential algebraic PDEs and bounds on the coefficients of Kolchin polynomials under projection in the PDE case. Now that we have obtained the former in Theorem 5.13 together with a bound for characteristic sets, it is possible to obtain the latter in Lemma 6.6 and finish the proof. Therefore, Section 6 can be thought of as a motivation for the rest of the paper and is an interesting example of a problem from differential-difference algebra that motivated a purely differential algebraic result.

6.1. Bounds for Kolchin polynomials for algebraic PDEs

Definition 6.1.

Let be a differentially closed -field containing a -field . We say that is a -variety over if there exists such that

We write . The -variety is denoted by . A -variety is called irreducible if the differential ideal is prime.

For a subset , the smallest -variety containing is called the Kolchin closure of and denoted by .

Definition 6.2.

We will say that a -variety is bounded by if () and can be defined by equations of order and degree at most .

Notation 6.3.

For a numeric polynomial , we set

Definition 6.4.

The generic point of an irreducible -variety , where , is the image of under the homomorphism .

Definition 6.5.

The Kolchin polynomial of an irreducible -variety , where , is the unique numerical polynomial such that there exists such that, for all and the generic point of , , where . For the proof of the existence, see [22, Theorem 5.4.1].

Lemma 6.6.

There exists a computable function such that for every

  • differential variety bounded by ,

  • irreducible component ,

  • and linear projection ,

we have , where .

Proof.

By performing a linear change of variables, we reduce the problem to the case in which is the projection to the first coordinates. Consider a ranking such that

  • is greater than every derivative of for every and ;

  • the restriction of the ranking on is an orderly ranking (that is, a ranking such that implies, for all and , ).

Theorem 5.13 implies that has a characteristic set with respect to this ranking with the order bounded by a computable function of . Since a characteristic set of can be obtained from by selecting the polynomials only in the first variables, there is a charactersitic set of with respect to the orderly ranking with the order bounded by a computable function of . Then [15, Proposition 3.1] and [15, Fact 2.1] imply that is bounded by a computable function of . ∎

Proposition 6.7.

There exists an algorithm that, for every computable function , produces a number such that, for every sequence of Kolchin polynomials

such that for every , we have .

Proof.

By replacing with , we can further assume that is increasing and . [22, Definition 2.4.9 and Lemma 2.4.12] define a computable order-preserving map from the set of all Kolchin polynomials to (considered with respect to the lexicographic ordering). For , we define . For every function , we define

Note that if was computable, then is also computable.

The sequence gives rise to a sequence in with for every . [20, Main Lemma] implies that there is an algorithm to compute the maximal length of such a sequence, so there is an algorithm to compute a bound on from . ∎

6.2. Trains of varieties, partial solutions, and their upper bounds

Lemma 6.8.

For every --field of characteristic zero, there exists an extension of --fields, where is a differentially closed --field.

Proof.

The proof follows [18, Lemma 6.1] mutatis mutandis as follows. We will show that there exists a --field containing . The proof of [17, Proposition 2.1.7] implies that one can build an ascending chain of -fields

(2)

such that, for every , there exists an isomorphism of -fields, , and for every . We transfer the --structure from to ’s via ’s. Then implies that the restriction of on to coincides with the action of on . We set . Since the action and is consistent with the ascending chain (2), is a --extension of . It is shown in [17, Proposition 2.1.7] that the action of