Effective characterization of evaluation ideals of the ring of integro-differential operators

This paper provides a step forward to developing an algorithmic study of linear systems of polynomial ordinary integro-differential equations over a field <Formula format="inline"><TexMath><?TeX k?></TexMath><AltText>Math 1</AltText><File name="issac24-12-inline1" type="svg"/></Formula> of characteristic zero. Such a study can be achieved by first obtaining a constructive proof of the coherence property of the ring <Formula format="inline"><TexMath><?TeX I₁(k)?></TexMath><AltText>Math 2</AltText><File name="issac24-12-inline2" type="svg"/></Formula> of linear ordinary integro-differential operators with coefficients in <Formula format="inline"><TexMath><?TeX k[t]?></TexMath><AltText>Math 3</AltText><File name="issac24-12-inline3" type="svg"/></Formula>. To do that, the finiteness of the intersection of two finitely generated ideals has to be algorithmically studied. Three cases must be considered: first when evaluation operators generate the two ideals; second, when only one ideal is generated by evaluation operators; and third, when none is generated by evaluation operators. In this paper, we first explicitly characterize the intersection of two finitely generated ideals defined by evaluation operators. As for the second case, a key result is that the ideals generated by evaluations are semisimple <Formula format="inline"><TexMath><?TeX I₁?></TexMath><AltText>Math 4</AltText><File name="issac24-12-inline4" type="svg"/></Formula>-modules. We develop an algorithmic proof of this result. In particular, we show how a finite set of generators, defined by "simple" evaluations, can be obtained, that characterizes the class of finitely generated evaluation ideals of <Formula format="inline"><TexMath><?TeX I₁?></TexMath><AltText>Math 5</AltText><File name="issac24-12-inline5" type="svg"/></Formula> as finitely generated <Formula format="inline"><TexMath><?TeX k[t]?></TexMath><AltText>Math 6</AltText><File name="issac24-12-inline6" type="svg"/></Formula>-modules. Due to lack of space, the second and third cases will be developed in other publications.