Exercise 10, § 4

On utilise les notations des deux exercices précédents. Soient $l \in \mathbf{N}$ et $A \subset \mathscr{M}_l$. Pour tout couple $(i, d)$ on pose

et on note $C_{(i;d)}A$ the set of the $\mathrm{Card}(A_{(i;d)})$ first elements of $(\mathscr{M}_l)_{(i;d)}$ (for the lexicographic order). On pose $C_iA = \bigcup_{d=1}^l C_{(i;d)}A$. a) Pour tout $m \in \mathscr{M}_l$, on note $\pi(m)$ the predecessor in $\mathscr{M}_l$ of $m$ for the inverse lexicographic order, when it exists (i.e. $m \neq X_1^l$). Soient $i$ un entier tel que $1 < i$ et $a_i$ un entier strictly positive. Prove that one has b) Soient $A \subset \mathscr{M}_l$ et $p \geq 4$ tels que l’on ait $A \subset [X_p^l]$ (i.e. tel que les monomials belonging to $A$ involve only the variables $X_j, 1 \leq j \leq p$). On suppose que pour tout entier $i$, tel que $1 \leq i \leq p$, on a $A = C_iA$. Prove that $A = CA$ (notations de l’exerc. 9, d)). c) Soit $A \subset \mathscr{M}_l$ tel que $A \subset [X_3^l]$ et que $A = C_iA$ pour $i = 1, 2, 3$. Posons $A = \bigcup_{k=0}^l B_kX_3^k$ avec $B_k \subset [X_2^{l-k}]$ et $b(k) = \mathrm{Card}(B_k)$. Prove that $B_k$ is an initial subset of $\mathscr{M}_{l-k}$, that one has $0 \leq b(0) \leq l + 1, b(k+1) < b(k)$ if $b(k) \neq 0$, and that $k \mapsto b(k)$ is decreasing. Examine the converse of this assertion. Let $\mu$ be the number of the integers $k$ such that $b(k) = l + 1 - k$. Prove that $\mathrm{Card}(dA) = (\mathrm{Card}(A)) - \mu$.