Further results on the computation of the annihilators of integro-differential operators

This paper exposes some effective aspects of the algebra of linear ordinary integro-differential operators with polynomial coefficients. More precisely, we prove that the annihilator of an evaluation operator is a finitely generated ideal which can be explicitly characterized and computed. This is an advance towards the development of an effective elimination theory for ordinary integro-differential operators and an effective study of linear systems of integro-differential equations with polynomial coefficients.


INTRODUCTION AND MOTIVATION
Algebraic analysis, more specifically, algebraic -module theory, where  stands for "differential", is a mathematical field which studies linear systems of ordinary or partial differential equations using algebraic theories such as ring theory of differential operators, module theory, homological algebra [2,6,9].The main idea of this theory is to use a correspondence between linear systems of differential equations and finitely presented left modules over a ring of differential operators (e.g., the Weyl algebra of differential operators with polynomial coefficients).This theory is nowadays well-known in fundamental mathematics.In the last decades, the development of an effective approach to algebraic -module theory was studied by the computer algebra community.It relies on an effective differential elimination theory using, e.g., Gröbner or Janet basis methods.Different softwares can nowadays handle effective aspects of algebraic analysis: Macaulay2, Maple (OreModules), Singular, HomAlg, etc.
Based on our experience in effective algebraic -modules theory, we aim at extending the algebraic analysis approach to handle linear systems of ordinary integro-differential equations with polynomial coefficients.In other words, we would like to replace the Weyl algebra by the ring of ordinary integro-differential operators with polynomial coefficients, denoted by I 1 in what follows.Contrary to the Weyl algebra case, Bavula proved in [1] that I 1 is not a noetherian ring, a fact which seems to compromise the possibility to develop an effective integro-differential elimination theory, and thus, an effective algebraic analysis approach to ordinary integro-differential linear systems.However, he also proved that I 1 is coherent [1], namely, that every finitely generated left/right ideal of I 1 is finitely presented [2,18,20].As explained below, this result is at the core of the future development of an effective integro-differential theory.Yet, Bavula's proof of the coherence of I 1 remains not algorithmic.
This paper aims at effectively characterizing the annihilator of an integro-differential operator with polynomial coefficients, i.e., of an element of I 1 .In [17], such an effective characterization was obtained and implemented for an element of I 1 which is not a socalled evaluation operator.In this paper, we handle the second case, namely, the case of an element of I 1 which is an evaluation operator.This result completes the algorithmic characterization of the first of the two standard conditions characterizing the coherence property of I 1 .The second one asserts that the intersection of two finitely generated left/right ideals is also finitely generated.This problem will be studied in a future publication.
To further motivate this work, let us explain why the development of an effective version of the coherence property of I 1 plays a central role towards an effective study of linear systems of integro-differential equations with polynomial coefficients and towards the development of dedicated implementations built upon modern computer algebra systems.Within the algebraic analysis approach, a linear system of integro-differential equations with polynomial coefficients is defined by a matrix  ∈ I × 1 , i.e., by   = 0, where  ∈ F  ×1 and F is a left I 1 -module (e.g., k[],  ∞ (R)).It can be proved that the linear integro-differential system   = 0 is associated with the finitely presented left I 1 -module .Hence, the theory of linear integro-differential systems deals with the category of finitely presented left I 1 -modules [2,9].Now, a standard result in module theory [2,18,20] asserts that if R is a left coherent ring, then a left R-module M is coherent (namely, M is a finitely generated left R-module and all of its finitely generated left R-submodules are finitely presented) if and only if M is finitely presented.Combining this result with the fact that I 1 is coherent, we obtain that the finitely presented left I 1 -module M = coker I 1 (.) − associated with the linear system   = 0 − is coherent.In other words, due to the coherence property of I 1 , the linear system theory over I 1 deals with the study of the category of left coherent I 1 -modules.Now, standard theorems on finitely generated modules over noetherian rings can be extended to finitely presented modules over coherent rings.Moreover, the coherence property is compatible with all the standard algebraic operations (e.g., direct sum, intersection, quotient, tensor product, homomorphism, kernel, image, cokernel).For more details, see [2,18,20].Therefore, if the coherence property of I 1 is made algorithmic and implemented in computer algebra systems, then the algebraic side of linear system theory over I 1 can also be made effective.Note that, to our knowledge, I 1 would be the first example of a coherent but not noetherian ring implemented in a computer algebra system, that has important applications (e.g., calculus).

THE RING OF INTEGRO-DIFFERENTIAL OPERATORS
In what follows, let k be an algebraically closed field of characteristic 0 (e.g., and the k-linear endomorphisms defined on the basis (  ) ∈N of k[] as follows: These k-endomorphisms respectively define the following linear operators acting on k[]:

Definition 2.1 ([1]
).A 1 is the k-subalgebra of E generated by  and , and I 1 is the k-subalgebra of E generated by , , and  .
Then, A 1 (k), or simply A 1 , is called the Weyl algebra defining the ordinary differential operators with polynomial coefficients.Similarly, I 1 (k), or simply I 1 , is the ring of ordinary integro-differential operators in the variable  with polynomial coefficients in k[].
In particular, we have the inclusion A 1 ⊂ I 1 .The first fundamental theorem of calculus can be rewritten as   = 1, where 1 stands for the identity of E.Moreover, we can see that for every  ∈ k[], (1 −  ) () =  ( 0 ), which shows that the operator belongs to I 1 .We shall refer to it as the evaluation operator.Note that  is multiplicative, i.e.,  ( The second fundamental theorem of calculus then rewrites   = 1 −.
In what follows, we shall simply set  0 to 0.
Contrary to its subring A 1 , the ring I 1 has nontrivial zero divisors since   = 0 and   = 0.
For  ∈ k[], we have the following fundamental identities in I 1 : See, e.g., [15].We can deduce the following extra identities: =  ()  −   (),    =  () .Note that the first of the aforementioned identities corresponds to the Leibniz rule and the last but one to the integration by parts.
We state again that an element  of I 1 can be written uniquely as where   ,   ,   ,   ∈ k[] and , ,  ∈ N.For more details, see [1,10,17].The identity (2) is called the normal form of .
For instance, let us give the explicit normal form of   .First, setting  = 1 in the identity  =  ()  −   () and using  (1) = , we obtain the identity  2 =   −  , which corresponds to the double integration.More generally, we have the following explicit result on multiple integrations (that does not seem to appear in the literature).
Proposition 2.2.The operator   can be written as a polynomial of degree 1 in I.More precisely, we have Let  be a nonnegative integer.On the one hand, we have On the other hand, we have To conclude, we thus have to prove the following identity: To do so, let us note  = and compute its partial fraction expansion, i.e., write it as ! (− −1)! , which proves (5) and thus (3).□ Note that Formula (3) holds for any  0 ∈ k in the definition of  (see (1)) and not only for  0 = 0.

Proposition 2.3 ([1]
).The set ⟨⟩ = I 1  I 1 is the only two-sided ideal of I 1 .Moreover, we have: where where  () stands for the application of (1) to .Note that Let us state a result that will be useful in what follows.
Proof.Let us consider  ∈ I 1 .Then, there are  1 ∈ A 1 ,  2 ∈ I, and  3 ∈ ⟨⟩ such that  =  1 + 2 + 3 .By linearity and associativity properties, we only have to prove that    =  () , for all  ∈ k[].If  1 =  =0     , then, using the identity   = 0, we get If  2 =  =0      , then, using the identity    =  () , for all  ∈ k[], we have and  2 = , we obtain Thus, we have    =  () , for all  ∈ k[].This ends the proof.□ We state again that the annihilator of  ∈ I 1 is defined by By Lemma 2.4, the annihilator of  =  =0      ∈ ⟨⟩ can be described by ann It is well-known that A 1 is a noetherian ring, i.e., is a left and a right noetherian ring (see, e.g., [2,6]).As for I 1 , the situation is different.Indeed, we have seen that the identities   = 1 and   = 1 −  hold in I 1 .Now, a theorem due to Jacobson [8] asserts that the existence of a left/right inverse, which is not a two-sided inverse, of an element in a noncommutative ring R implies that R is not left/right noetherian.We thus have the following result.Proposition 2.5 ( [1]).The ring I 1 is neither a left nor a right noetherian ring.
A more explicit proof of Proposition 2.5 consists in considering the chain of left ideals generated by the Taylor operators defined by One can check that    +1 =   and I 1   ≠ I 1  +1 , for all  ∈ N, so that

Proposition 2.6 ([1]
).I 1 admits the involution  defined by An important consequence of ( 7) is that  () =  and  (⟨⟩) = ⟨⟩.Moreover, the involution  defined in Proposition 2.6 can be used to turn the strictly ascending chain of left ideals exhibited above into a strictly ascending chain of right ideals of I 1 .This proves that I 1 is also not a right noetherian ring.
At first sight, the fact that I 1 is not a noetherian ring seems to be a strong obstruction to a pure algebraic, and thus to an effective, study of I 1 .The next section explains why an effective study of linear systems of integro-differential equations with polynomial coefficients remains feasible.

THE COHERENCE PROPERTY OF I 1
In this section, let R be a ring and M a left R-module.Definition 3.1.A left R-module M is said to be finitely generated if there is a finite family A left R-module M is finitely generated if there exist some surjective R-homomorphism  : R 1× −→ M, i.e., a R-epimorphism.Let   = (0, . . ., 1, . . ., 0) be the  ℎ element of the standard basis of R 1× , namely, the row vector of length  with 1 at the  ℎ position and 0 elsewhere.If (  )  ∈ 1, is a set of generators of M, then we can consider the following R-epimorphism: Definition 3.2.Let M be a finitely generated left R-module and (  )  ∈ 1, a finite set of generators of M.Then, M is said to be finitely presented if the left R-module of all the relations among the   's, namely, By definition, a finitely presented module is finitely generated.The fact that ker  is finitely generated is equivalent to the existence of a finite set of generators of ker , i.e., a finite set of elements  1• , . . .,  • ∈ R 1× satisfying that, for all  ∈ ker , there are  1 , . . .,   ∈ R such that Thus, we can write ker  = im R (•) = {  |  ∈ R 1× }, where  ∈ R × is the matrix having the  • 's as rows, which is equivalent to the following exact sequence of left R-modules: , where means "isomorphic to".For more details, see [2,18,20].
Let R be a noncommutative ring.
• A left R-module M is said to be left coherent if M is a finitely generated left R-module and if every finitely generated left R-submodule of M is finitely presented.• The ring R is said to be left coherent if R is a left coherent R-module, i.e., if every finitely generated left ideal of R is finitely presented.Similar definitions hold for right R-modules and a ring is said to be coherent if it is both left and right coherent.
According to Definitions 3.3 and 3.2, a ring R is left coherent if for every finitely generated ideal J of R, the left R-module of the relations among a finite set of generators of J is finitely generated.Let us state a useful characterization of a coherent ring.Proposition 3.5 ([18, 20]).Let R be a ring.The following two conditions are equivalent: (1) R is a left coherent ring.
(b) For all finitely generated left ideals J 1 and J 2 , the left ideal J 1 ∩ J 2 is finitely generated.
A similar result holds for a right coherent ring (ann R (.) is then replaced by ann R (.) and left ideals by right ideals).
We can now state a result that is at the core of this paper.
Using the involution  of I 1 (see Proposition 2.6), the left coherence property yields the right coherence property, and vice versa.
We point out that the proof of Theorem 3.6 given in [1] is not constructive.The main goal of the present paper is to contribute to the development of an effective version of the coherence property of I 1 and its implementation in the computer algebra software Maple.Here, we shall focus on Condition (2)(a) of Proposition 3.5.The second condition will be studied in a future work.
Note that, to our knowledge, I 1 would be the first example of a coherent but not noetherian ring implemented in a computer algebra system, which has important applications (e.g., calculus).

ANNIHILATOR OF AN EVALUATION OPERATOR 4.1 Preliminary remarks and results
In Section 3, we have recalled that the ring I 1 was coherent.Yet, the proof of the coherence property given in [1] remains not algorithmic.To make it so, we shall rely on Proposition 3.5 which shows that the coherence property is equivalent to two conditions: one on the annihilator of elements of I 1 and one on the intersection of finitely generated ideals.In this paper, we shall only focus on the first one, letting the second one for a future work.
Let us then consider Condition (2) (a) of Proposition 3.5.In the case  ∈ I 1 \ ⟨⟩, the characterization of a finite set of generators for ann I 1 (.) was obtained in [17] and implemented in the IntDiffOp package [10].Hence, it remains to study the case  ∈ ⟨⟩.
Note that ann I 1 (.) is the left ideal of I 1 defining all the compatibility conditions of the inhomogeneous linear equation  ℎ = , where  is fixed in a left I 1 -module F and ℎ is sought in F .Indeed, if  ∈ ann I 1 (.), then by definition,   = 0, i.e.,   = 0.This last equation is a necessary condition for the above system to have a solution, i.e.,   = 0 is a compatibility condition.
We now state two lemmas that will be used in what follows.
Proof.Using (1), let us compare (    )   and (1 −  −1 )   , for all  ≥ 0. If  ≤  − 1, we have Using (4), we obtain Then, we have The second assertion is a direct consequence of the first one, Lemma 4.1, and Proposition 2.3.□ Using Proposition 2.2, for all  ≥ 1 and  ∈ k[], we have Hence, the identity for all  ≥ 1.This shows that Lemma 4.2 encapsulates the Taylor's theorem with an integral form of the remainder into a simple identity.

Annihilator of a simple evaluation operator
Let  ∈ k[] and let us exhibit a generating set of the left I 1 -ideal ann I 1 (. ).From Lemma 2.4, we have ann is the left ideal of A 1 generated by  1 :=  +1 and  2 :=    −  () .In other words, we have be a polynomial of degree ,  1 =  +1 , and  2 =    −  () .Then, we have
be a full row rank matrix, where  ∈ 1,  + 2 is the rank of  such that and the vector , where  >  + 1 and   ∈ k[], for  = 0, . . ., , then we can write and we get  =  =0     ∈  =0 K A 1 (  ).To simplify the exposition below, we keep the term of order  + 1 in .We then have: where the matrix  ∈ k[] (+2) × ( +1) is defined by (9).This shows that ( 0 . . . +1 ) ∈ ker k[ ] (.).Now, since k[] is a noetherian ring, the k[]-module ker k[ ] (.) is finitely generated (see, e.g., [7]), and thus, there is a finite set of generators for ker k[ ] (.).Stacking the corresponding polynomial rows into a matrix, we obtain ) , and thus, ker k[ ] (.) is a free k[]-module because k[] is a principal ideal domain (see, e.g., [18]).Hence, the rows of the matrix  can be chosen so that they are k[]-linearly independent, i.e., such that the matrix  has full row rank (i.e.,   = 0 yields  = 0).Therefore, we have the following exact sequence of k[]-modules: =1 A 1   , which proves the first inclusion.For the converse, we note that each entry of the column vector   +1 is an element of A 1 which annihilates all the   's because the rows of  belong to ker k[ ] (.).Hence, we obtain that ( 1 . . .  )  =   +1 is a generating set of  =0 K A 1 (  ).□ Note that the generators   's of Lemma 4.5 can be explicitly obtained by the computation of a Hermite normal form (or a Smith normal form) of the matrix .For more details, see, e.g., [11].Such a computation is implemented in computer algebra softwares.
We recall that a finitely generated R-module M is said to be projective if there are a R-module P and  ∈ N such that we have M ⊕ P R  .If R is an integral domain, the rank of a R-module M is the dimension of the  (R)-vector space  (R) ⊗ R M obtained by extending the coefficients of M from R to its quotient field  (R) = { 1 / 2 | 0 ≠  2 ,  1 ∈ R}.See [7,18].Let us now state a standard result on Fitting ideals.Theorem 4.14 ([7], Prop.20.8, p. 495).Let R be a commutative ring, M a finitely presented left R-module, and  ∈ R × a presentation matrix of M. The following assertions are equivalent: (1) M is a projective module of rank  .scope of the present paper.For the case of A 1 , see [16] and the references therein.

Example 3 . 4 .
Left (resp., right) noetherian rings are left (resp., right) coherent rings.Two examples of coherent rings which are not noetherian are the ring k[  |  ∈ N] of polynomials in an infinite number of variables {  }  ∈N with coefficients in a field k, and the ring of the entire functions on C. For more details, see[18].