6.3 Application au calcul d’un point par composante connexe dans un ensemble semi-alg´ebri-
6.4.3 Complexit´e et performances pratiques
On peut maintenant donner la complexit´e des versions probabilistes de l’algorithme d´ecrit ci-dessus lorsque la routine d’´elimination alg´ebrique utilis´ee est l’algorithme de r´esolution g´eom´etrique. L’entr´ee de l’algorithme est un syst`eme d’´equations et d’in´egalit´es polynomiales
f1=· · ·=fs= 0, g1>0, . . . , gk>0
et on noteraLla longueur d’un programme d’´evaluation de la famille (f1, . . . , fs, g1, . . . , gk).
Si dest la dimension de la vari´et´e alg´ebriqueV ⊂Cn d´efinie par : f1=· · ·=fs= 0
on peut (grˆace au th´eor`eme 46) borner le nombre de vari´et´es alg´ebriquesV(o`uI ⊂ {1, . . . , k}) `a ´etudier parPmin(d,k)
`=0
¡k
`
¢.
Pour chacune vari´et´e ag´ebrique V, on doit :
– calculer les valeurs critiques g´en´eralis´ees de la projection surεrestreinte `aV, la complexit´e de cette
´etape est donn´ee par le th´eor`eme 42 ;
– choisir un rationnel compris entre 0 et la plus petite de ces valeurs, la complexit´e de cette ´etape n’influe pas sur la complexit´e globale de l’algorithme ;
– calculer au moins un point par composante de Ve,aI ∩Rn, la complexit´e de cette ´etape est donn´ee par le th´eor`eme 29.
– d´eterminer le signe desgj(pourj∈ {1, . . . , k}\I) en les points r´eels encod´es par les param`trisations rationnelles fournies par l’´etape pr´ec´edente, la complexit´e de cette ´etape n’influe pas sur la com-pexit´e globale de l’algorithme.
Cette discussion permet donc d’´enoncer le r´esultat ci-dessous.
Th´eor`eme 47. SoitS ⊂Rn un ensemble semi-alg´ebrique d´efini par f1=· · ·=fs= 0, g1>0, . . . , gk >0
o`u(f1, . . . , fs, g1, . . . , gk)sont des polynˆomes deQ[X1, . . . , Xn] de degr´e born´e parD. SoitLla longueur d’un programme d’´evaluation de(f1, . . . , fs, g1, . . . , gk).
Si la vari´et´e alg´ebrique d´efinie par f1 = · · · =fs = 0 est lisse et que l’id´eal hf1, . . . , fsi est radical et ´equi-dimensionnel de dimension d, l’algorithme ci-dessus calcule au moins un point par composante connexe deS en
O(n7(n−d)4dD4n) op´erations arithm´etiques dansQ.
Implantations et performances pratiques. De premi`eres implantations de cet algorithme utilisant les bases de Gr¨obner comme routine d’´elimination alg´ebrique ont ´et´e effectu´ees. Bien ´evidemment, l’usage des bases de Gr¨obner permet d’obtenir des versions d´eterministes de l’algorithme qu’on vient de d´ecrire.
Cet algorithme tirant profit de l’efficacit´e des algorithmes donn´es dans le chapitre pr´ec´edent sur les probl`emes de plus de 4 variables, cette premi`ere implantation a d’ores et d´ej`a permis de r´esoudre des applications inaccessibles `a l’algorithme de d´ecomposition cylindrique alg´ebrique. Ceci dit, il apparaˆıt que diverses strat´egies peuvent ˆetre employ´ees (dans l’ordre d’´etude des famillesI) et donnent des r´esultats pratiques sensiblement diff´erents : sur certains probl`emes certaines sont plutˆot lentes devant d’autres et le rapport d’inverse sur d’autres prob`emes. Il reste encore un certain nombre de choses `a comprendre et/ou un travail d’implantation `a effectuer avant d’arriver `a des r´esultats pratiques satisfaisants.
6.5 Notes bibliographiques et commentaires
Le calcul d’au moins un point par composante connexe d’un ensemble semi-alg´ebrique d´efini par un syst`eme d’´equations et d’in´egalit´es polynomiales par d´eformation pour ensuite appliquer les m´ethodes de points critiques apparaˆıt sous diff´erentes formes dans [65, 69, 70, 114, 17, 18, 19]. Ces algorithmes n’ont en g´en´eral aucune restriction sur l’entr´ee contrairement `a celui du paragraphe 6.4 de ce chapitre.
Ils souffrent n´eanmoins de constantes de complexit´e situ´ees en exposant particuli`erement ´elev´ees et sont inutilisables en pratique.
Les valeurs critiques g´en´eralis´ees sont initialement introduites dans [82, 74, 75]. Ces travaux four-nissent les premiers algorithmes permettant leur calcul via des id´eaux d’´elimination (l’un de ces algo-rithmes est ´evoqu´e dans le paragraphe 6.1). La taille des donn´ees interm´ediaires est particuli`erement
´elev´ee devant la taille de la sortie, si bien que ces algorithmes sont tr`es peu efficaces.
L’id´ee d’utiliser des propri´et´es de propret´e pour ramener le calcul de valeurs critiques asymptotiques d’une application polynomiale pour ramener leur calcul `a celui du lieu de non-propret´e d’une projection restreinte `a une courbe critique apparaˆıt initialement dans [129] et est d´evelopp´ee dans [131]. On y trouve les algorithmes et estimations de complexit´e donn´es dans le paragraphe 6.1. L’algorithme de calcul d’au moins un point par composante connexe d’un ensemble semi-alg´ebrique d´efini par une seule in´egalit´e ou une seule in´equation du paragraphe 6.3, ainsi que son application `a la d´etermination de l’existence de points r´eels r´eguliers dans un ensemble alg´ebrique d´efini par une seule ´equation apparaissent aussi dans [131].
Ce chapitre n’aborde pas le calcul d’au moins un point par composante connexe d’un ensemble semi-alg´ebrique d´efini par un syst`eme d’´equations et d’in´egalit´es (non strictes) polynomiales
f1=· · ·=fs= 0, g1>0, . . . , gk >0
Ce probl`eme se ram`ene en fait directement `a l’´etude des vari´et´es alg´ebriques r´eelles d´efinies par f1=· · ·=fs=gi1 =· · ·=gi` = 0
pour tout I = {i1, . . . , i`} ⊂ {1, . . . , s} (voir [21, Chapitre 13]). Une ´etude de ce probl`eme et son application `a un probl`eme de reconnaissance de forme figurent aussi dans [91].
Enfin, le calcul de valeurs critiques g´en´eralis´ees d’une application polynomiale restreinte `a une vari´et´e alg´ebrique lisse peut ˆetre utilis´e pour ´eviter les choix de projection g´en´erique (et/ou v´erifier que les choix de projection sont suffisamment g´en´eriques) dans les algorithmes du paragraphe 5.5 du chapitre pr´ec´edent. En effet, ces algorithmes sont fond´es sur le fait que les composantes connexes des vari´et´es choisies ont une image ferm´ee (pour la topologie euclidienne) par les projections choisies. Si ce n’est pas le cas, les extrˆemit´es de ces intervalles sont fatalement des valeurs critiques asymptotiques (puisque dans ce cas, on ne peut pas avoir fibration localement triviale autour ces extrˆemit´es). Un calcul de valeur critique asymptotique permet de r´ecup´erer ces valeurs et il suffit alors de consid´erer des fibres au-dessus de rationnels choisis entre chacune des valeurs critiques asymptotiques calcul´ees.
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