Structure combinatoire des idéaux monomiaux

In this Thesis, we study monomial ideals from a combinatorial point of view. We are mainly interested in the structure of the associated Groebner escalier but, sometimes, we have also to deal with the initial ideal. First of all, we examine all the existing combinatorial methods to compute the Groebner escalier N(I(X)) associated to the zerodimensional radical ideal I(X) of a finite set of distinct points X. More precisely, we start from Cerlienco-Mureddu correspondence and we examine the other methods which came up later on, such as Gao-Rodrigues-Stroomer method, Lederer’s variation and Lex Game. Next, we face the problem of constructing a linear factorization of a minimal Groebner basis for a zerodimensional radical ideal. The existence of such a factorization has been stated and proved by Maria Grazia Marinari and Teo Mora, in the Axis of Evil Theorem [2, 69, 70]. In this Thesis we give an alternative constructive proof, together with an algorithm computing concretely the factorization and we study deeply the structure of the Groebner escalier, in connection to the Axis of Evil factorization.Then, we develop a visual language in order to represent finite sets of terms and infinite order ideals via bidimensional pictures, the Bar Codes. We show that the pictures we get allow us to read easily many properties of the monomial ideal (expecially connected to Janet decomposition for terms [54, 55, 56, 57]) and to developan iterative version of the Axis of Evil algorithm.Thanks to the Bar Code structure, moreover, we are able to connect commutative algebra and enumerative combinatorics, by giving a bound for the number of strongly stable ideals with a fixed constant affine Hilbert polynomial, by putting them in biunivocal correspondence with plane partitions.Finally, we show how the Axis of Evil theorem can be applied to coding theory, more precisely to the decoding procedure for binary BCH codes and to the computation of sparse general error locator polynomials.Dans la thèse on étude les idéaux monomiaux dans un point de vue combinatoire, et on se concentre sur leur escalier de Groebner. Des nombreux liens avec la théorie de codes et la combinatoire énumérative sont fournis. De plus, le code de certains logiciels associés est joint à la thèse