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\def\pointir{\discretionary{.}{}{.\kern.35em---\kern.7em}\nobreak \hskip 0em plus .3em minus .4em } \def\resume#1{\vbox{\eightpoint \pc R\'ESUM\'E|\pointir #1}} \def\abstract#1{\vbox{\eightpoint \pc ABSTRACT|\pointir #1}} \def\titre#1|{\message{#1} \par\vskip 30pt plus 24pt minus 3pt\penalty -1000 \vskip 0pt plus -24pt minus 3pt\penalty -1000 \centerline{\bf #1} \vskip 5pt \penalty 10000 } \def\section#1|{\par\vskip .3cm {\bf #1}\pointir} \def\ssection#1|{\par\vskip .2cm {\it #1}\pointir} \long\def\th#1|#2\finth{\par\medskip {\petcap #1\pointir}{\it #2}\par\smallskip} \long\def\tha#1|#2\fintha{\par\medskip {\petcap #1.}\par\nobreak{\it #2}\par\smallskip} \def\cf{{\it cf}} \def\rem#1|{\par\medskip {{\it #1}\pointir}} \def\rema#1|{\par\medskip {{\it #1.}\par\nobreak }} % \def\ieme{\raise 1ex\hbox{\pc{}i\`eme|}} \def\omini{\raise 1ex\hbox{\pc{}o|}} \def\emini{\raise 1ex\hbox{\pc{}e|}} \def\ermini{\raise 1ex\hbox{\pc{}er|}} \def\remini{\raise 1ex\hbox{\pc{}re|}} %reference pour un article : \def\article#1|#2|#3|#4|#5|#6|#7| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3, {\sl #4}, t.\nobreak\ {\bf #5}, {\oldstyle #6}, p.\nobreak\ #7.\par}} %reference pour un livre : \def\livre#1|#2|#3|#4| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir {\sl #3}\pointir #4.\par}} %reference complementaire : \def\divers#1|#2|#3| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3.\par}} % \mathchardef\conj="0365 \def\dem{\par{\it D\'emonstration}\pointir} \def\qed{\quad\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width 4pt \vfill\hrule}\vrule}} \def\virg{\raise 2pt\hbox{,}} % virgule aprs une fraction \def\cqfd{\penalty 500 \hbox{\qed}\par\smallskip} \def\decale#1|{\par\noindent\hskip 28pt\llap{#1}\kern 5pt} \def\decaledecale#1|{\par\noindent\hskip 34pt\llap{#1}\kern 5pt} % pour les titres en deux lignes et les sections sans point-tiret : \def\titrea#1|#2|{\message{#1 #2} \par\vskip.5cm plus .1cm minus .1cm\penalty -1000 \centerline{\bf #1} \centerline{\bf #2} \vskip 5pt \penalty 10000 } \def\sectiona#1|{\par\vskip .3cm {\bf #1.} \par\nobreak\vskip 3pt } \def\ssectiona#1|{\par\vskip .2cm {\it #1.} \par\nobreak\vskip 2pt } \catcode`\@=12 \def\supp{\mathop{\rm supp}} \pageno=5 \auteurcourant={M. BARNABEI} \titrecourant={COMPOSITIONAL INVERSION} \eightpoint \leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1984$, 229/S--08} \leftline{Actes 8\emini\ S\'eminaire Lotharingien, p. 5--10} \tenpoint \let\it=\sl \vskip 1.5cm \centerline{{\bf COMPOSITIONAL INVERSION}} \vskip 5pt \centerline{{\bf OF TRIANGULAR SETS OF SERIES}} \vskip 2.8mm \centerline{\sevenrm BY} \vskip 2.8mm \centerline{{\petcap Marilena} BARNABEI} \vskip 5mm There are two possible definitions of formal Laurent series in infinitely many variables : they can be requested to be either series such that the set of their monomials has an infimum, or series for which the set of weights of their monomials has a minimum : in this second case, it is necessary to add a finiteness condition on monomials of the same weight, in order to perform products and, hence, compositions. So, we get two different algebras of Laurent series, say $L^1$, $L^2$, which are not comparable. $L^1$ has properties which are very similar to those of the algebra of Laurent series in one variable, and these properties have been extensively studied in [1]. On the contrary, in $L^2$ we find many pathological situations, including the fact that it seems impossible to characterize those sets of power series (namely, series whose monomials have only positive exponents) which can be composed with any Laurent series. Nevertheless, it is very natural to look for a Lagrange-type inversion formula which allows us to inverte such a ``simple'' set of series as $(x_i+x_{2i}^2)$, $i\in {\bf N}$. Here, we solve this problem for a special class of sets of power series in $L^2$, namely, triangular $P$-sets. Our result is completely characteristic-free. \vskip 5pt Let ${\bf D}$ be the set of all maps ${\bf d} :{\bf N}\rightarrow {\bf Z}$ with finite support ; such maps will be called {\it degrees}. Sum and order relation between degrees are defined pointwise. The {\it weight} of a degree ${\bf d}$ is the integer $w({\bf d}) :=\sum {\bf d}(i)$. Set $${\bf D}^+ :=\{ {\bf d}\in {\bf D} ; {\bf d}\geq {\bf 0}\} $$ where ${\bf 0}$ is the zero-degree. Let now ${\bf A}$ be a commutative integral domain with unity, and let ${\bf U}$ denote the group of units of ${\bf A}$. A {\it formal series in infinitely many variables} over ${\bf A}$ will be a map $\alpha :{\bf D}\rightarrow {\bf A}$. We set $a_{{\bf d}} :=\alpha ({\bf d})= : \langle {\bf d}\mid \alpha \rangle $ and write $\alpha =\sum _{{\bf d}}a_{{\bf d}}\tau ^{\bf d}$, where $\tau _1,\tau _2,\ldots$ are formal variables, and $\tau ^{{\bf d}} :=\prod _n\tau _n^{{\bf d}(n)}$. \vskip 5pt A series $\alpha $ will be called a {\it Laurent series} whenever : 1) for every $k\in {\bf Z}$, the set $\{ {\bf d}\in {\bf D} ; \langle {\bf d}\mid \alpha \rangle \not= 0\ {\rm and}\ w({\bf d})=k\} $ is finite, and 2) the set $\{ w({\bf d}) ;{\bf d}\in {\bf D},\langle {\bf d}\mid \alpha \rangle \not=0\} $ has a minimum, which is called the {\it weight} of $\alpha $ and will be denoted by $w(\alpha )$. The weight of the zero series is meant as $+\infty $. \vskip 5pt Let $\alpha ,\beta $ be two Laurent series ; the {\it convolution product} $\alpha \beta $ is defined as $$\alpha \beta :=\sum _{{\bf d}} \left(\sum _{{\bf f}}\langle {\bf f}\mid \alpha \rangle \langle {\bf d}-{\bf f}\mid \beta \rangle \right)\tau ^{{\bf d}} ; $$ this makes sense, since the inner sums are finite, and yields a Laurent series. It is immediate that, under the usual pointwise sum and the convolution product, the set $L$ of all Laurent series turns out to be a commutative ${\bf A}$-algebra. It can be also proved that $L$ is an integral domain, and for every $\alpha ,\beta \in L$, $w(\alpha ,\beta )=w(\alpha )+w(\beta )$. The identity element with respect to the product will be denoted by $\upsilon $. We explicitly note that the convolution product $\alpha \beta $ is defined even if $\alpha $ is a Laurent series and $\beta $ is any series satisfying condition~2). Straightforward computations show that a Laurent series $\alpha $ admits multiplicative inverse $\alpha ^{-1}$ whenever it has exactly one monomial of minimum weight $\beta $. If this is the case, setting $\alpha =\beta +\gamma $, we have $$\alpha ^{-1}=\beta ^{-1}\sum _{n\geq 0}(-1)^n \gamma ^n\beta ^{-n}.$$ A {\it power series} will be a Laurent series $\alpha $ such that $\langle {\bf d}\mid \alpha \rangle \not=0$ only for ${\bf d}\in {\bf D}^+$. The set $P$ of all power series is of course a subalgebra of $L$. A collection $\alpha :=(\alpha _i)$, $i\in {\bf N} $, of power series will be called a $P$-{\it set} whenever : i) $w(\alpha _i)>0$ for every i ; ii) for every degree ${\bf d}$, the set $\{ i\in {\bf N} ;\langle {\bf d}\mid \alpha _i\rangle \not=0\}$ is finite. If $\alpha :=(\alpha _i)$ is a $P$-set and ${\bf d}\in {\bf D}$, set $\alpha ^{{\bf d}} :=\sum _i\alpha _i^{{\bf d}(i)}$. \vskip 5pt Let now $\alpha :=(\alpha _i)$ be a $P$-set, and $\beta $ a Laurent series. The {\it composition} $\beta \circ \alpha $ is defined as $$\beta \circ \alpha :=\sum _{{\bf d}}\langle {\bf d}\mid \beta \rangle \alpha ^{{\bf d}}.$$ It is easily checked that the definition makes sense and, moreover : 1) if $\beta $ is a power series, $\beta \circ \alpha $ is again a power series ; 2) if $\beta $ is a Laurent series, $\beta \circ \alpha $ is a series with minimum weight (namely, a series satisfying condition 2)), but not, in general, a Laurent series : as an example, take $$\beta =\sum _{n\geq 2}\tau _n^{2n-2}/\tau _1^{n-1}$$ and $\alpha =(\alpha _i)$, with $\alpha _1=\tau _1^2$ and $\alpha _n=\tau _n$ for $n\geq 2$ ; then $$\beta \circ \alpha =\sum _{n\geq 2} (\tau _n/\tau _1)^{2(n-1)}$$ and this is a series consisting of infinite monomials of weight zero. In the sequel, we will consider only compositions between power series and $P$-sets. \vskip 5pt The {\it composition of two $P$-sets} $\alpha :=(\alpha _i)$ and $\beta $ is defined as follows : $$\alpha \circ \beta :=(\alpha _i\circ \beta ).$$ It is immediately seen that this gives an associative operation between $P$-sets, whose identity element is the $P$-set $\tau :=(\tau _i)$. \vskip 5pt We are now interested in those $P$-sets $\alpha $ which are invertible with respect to composition, namely, such that there exists another $P$-set $\beta $ such that $\alpha \circ \beta =\tau =\beta \circ \alpha $. If this is the case, then $\beta $ is of course unique, and we call it the {\it inverse set} of $\alpha $, denoting it by $\tilde\alpha $. It is obvious that, if $\alpha :=(\alpha _i)$ is an invertible $P$-set, then we must have $w(\alpha _i)=1$ for every i. In order to study invertible $P$-sets, we begin with the following special case : \th Proposition 2.1|Let $\alpha :=(\alpha _i)$ be a $P$-set, with $$\alpha _i=a_i\tau _i+\hat \alpha _i,\quad a_i\in {\bf U},\quad w(\hat \alpha _i)\geq 2,\quad i\in {\bf N}.$$ Then, $\alpha $ is invertible. \finth Obviously, the same result holds also for $P$-sets $\alpha :=(\alpha _i)$ of the form $$\alpha _i=a_i\tau _{\sigma (i)}+\hat \alpha _i\quad \quad (a_i\in {\bf U},\ w(\hat \alpha _i)\geq 2),$$ for some bijection $\sigma :{\bf N}\rightarrow {\bf N}$. Let now $\alpha :=(\alpha _i)$ be a $P$-set with $w(\alpha _i)=1$ for every i ; the {\it linear part} of $\alpha $ will be the $P$-set $\lambda :=(\lambda _i)$, with $$\lambda _i :=\sum _{{\bf d} :w({\bf d})=1} \langle {\bf d}\mid \alpha _i\rangle \tau ^{{\bf d}} \qquad {\rm for\ every}\ i.$$ A $P$-set which coincides with its linear part will be called a {\it linear} $P$-{\it set}. There exist linear $P$-sets whose compositional inverse consists of series not satisfying condition 1) (so, they are not power series) : as an example, take $\alpha _i :=\tau _i+\tau _{i+1},\quad i\in {\bf N}$. Then, it is easily checked that the set of series $$\beta _i :=\sum _{k\geq i}(-1)^{k+i}\tau _k,\quad i\in {\bf N}$$ is the compositional inverse of $(\alpha _i)$, and each series $\beta _i$ consists of infinitely many monomials of weight one. On the other hand, we have : \th Proposition 2|A $P$-set consisting of series of weight one is invertible if and only if its linear part is invertible. \finth In the sequel, we will be concerned with sets of power series $\alpha :=(\alpha _i)$, $i\in {\bf N}$, such that : 1) $\alpha _i=a_i\tau _i+\hat \alpha _i$, with $a_i\in {\bf U}$ and $w(\hat\alpha _i)\geq 2$ ; 2) $\partial \alpha _i/\partial \tau _j=0$ for $i>j$, where $\partial /\partial \tau _j$ denotes the (formal) partial derivative with respect to $\tau _j$. Such sets of series are automatically $P$-sets, and we will call them {\it triangular} $P$-{\it sets}. By \pd PROPOSITION 2.1, every triangular $P$-set is invertible. If $\alpha :=(\alpha _i)$ is a triangular $P$-set, set $$P(\alpha _i) :={\tau _i\over \alpha _i} {\partial \alpha _i\over \partial \tau _i}\qquad {\rm for\ every}\ i\; ;$$ $P(\alpha _i)$ is a Laurent series of weight zero, $\langle 0\mid P(\alpha _i)\rangle =1$, and in its monomials $\tau _i$ is the only variable which can appear with negative exponent. Moreover, by definition, $\langle {\bf d}\mid P(\alpha _i)\rangle \not=0$ implies $i\leq \min \supp({\bf d})$. Hence, setting $P(\alpha _i)=\upsilon +\gamma _i$, we have $\langle {\bf 0}\mid \gamma _i\rangle =0$ and, for every $J\subseteq {\bf N}$, $J$ finite, $\langle {\bf d}\mid \prod _{j\in J}\gamma _i\rangle \not=0$ only if $j\leq \max \supp({\bf d})$ for every $j\in J$. This implies that the infinite product $$\prod _{i\in {\bf N}}P(\alpha _i) :=\upsilon +\sum _{\emptyset \not=J\subseteq {\bf N}} \prod _{j\in J}\gamma _j\qquad (J\ {\rm finite})$$ is well-defined. We set $$P(\alpha ) :=\prod _{i\in {\bf N}}P(\alpha _i).$$ By previous remarks, for every ${\bf d}\in {\bf D}$, we have : $$\langle {\bf d}\mid P(\alpha )\rangle =\langle {\bf d}\mid \prod _{i\leq a}P(\alpha _i)\rangle $$ with $a :=\max \supp({\bf d})$. We note explicitly that $P(\alpha )$ is a series, but not, in general, a Laurent series : for example, if $\alpha _i :=\tau _i(\upsilon +\tau _j)^{-1}$, we have $P(\alpha _i)=(\upsilon +\tau _i)^{-1}$, and $P(\alpha )$ has infinitely many monomials of given weight. Nevertheless, $P(\alpha )$ can be multiplied by any Laurent series, since it has a minimum weight. \th Lemma|Let $\alpha :=(\alpha _i)$ be a triangular $P$-set ; then, for every ${\bf d}\in {\bf D}$, $$\langle {\bf 0}\mid \alpha ^{{\bf d}}P(\alpha )\rangle =\delta _{{\bf 0},{\bf d}}.$$ \finth \th Theorem 2 ({\rm Lagrange inversion formula})|Let $\alpha :(\alpha _i)$ be a triangular $P$-set, and $\tilde \alpha :=(\tilde \alpha _i)$ its compositional inverse ; then, for every ${\bf d}\in {\bf D}$, $$\langle {\bf d}\mid \tilde \alpha _i\rangle =\langle -{\bf e}_i\mid \alpha ^{-{\bf d}}P(\alpha )\rangle .$$ In particular, $\tilde \alpha $ is again a triangular $P$-set. \finth In [2], A. \pc JOYAL|, defines a composition between two power series in infinitely many variables, associating to any power series a set of series in a canonical way. As an example, we will translate this definition in our terminology, and derive an inversion formula for such sets of series. For any fixed $n\in {\bf N}$ let $\tau (n)$ be the $P$-set $$\tau (n) :=(\tau _n,\tau _{2n},\ldots\,).$$ It is evident that $\tau (k)\circ \tau (n)=\tau (kn)$ for every $k,n\in {\bf N}$. Let now $\alpha $ be any power series ; the {\it canonical set} associated to $\alpha $ will be the set $(\alpha \circ \tau (n))$, $n\in {\bf N}$. It is immediately seen that the canonical set associated to a series $\alpha $ is a $P$-set if and only if $w(\alpha )\rangle 0$. Moreover, if a canonical set $\alpha $ is invertible, its inverse is again a canonical set : \th Proposition 5.1|Let $\alpha $ be a power series whose canonical set $\alpha $ admits the inverse $\beta :=(\beta _n)$ ; then, $\beta $ is the canonical set associated to~$\beta _1$. \finth \th Theorem 3|Let $\alpha =a\tau _1+\hat \alpha $, $a\in {\bf U}$ and $w(\hat \alpha )\geq 2$, be a power series with canonical set $\alpha $, and let $\beta $ be the power series whose canonical set is the compositional inverse of $\alpha $ ; then $$\langle {\bf d}\mid \beta \rangle = \langle -{\bf e}_1\mid \prod _{k\geq 1}\bigl(\alpha ^{-{\bf d}(k)} P(\alpha )\bigr)\circ \tau (k)\rangle $$ where $$P(\alpha ) :={\tau _1\over \alpha }{\partial \alpha \over \partial \tau _1}. $$ \finth \vfill\eject \vglue 44pt \eightpoint \centerline{REFERENCES} \nobreak \vskip 10pt \divers 1|\pd BARNABEI (M.), \pd BRINI (A.) and \pd NICOLETTI (G.)|A General Umbral Calculus in infinitely many variables, to appear in {\sl Advances in Math}| \article 2|\pd JOYAL (A.)|Une th\'eorie combinatoire des s\'eries formelles|Advances in Math.|42|1981|1--82| \divers 3|\pd BARNABEI (M.)|A Lagrange inversion formula for triangular sets of series, to appear| \tenpoint \adresse {\petcap {\rm Marilena} Barnabei}, Dipartimento di matematica, Piazza di Porta San Donato 5, Universit\`a di Bologna, I-40127 Bologna, Italie \fin \bye