A proof of the dmenson conjecture and of Jacob's bound Franços Ollver (CNRS) LIX UMR CNRS École polytechnque
Jacob s bound Proposton. Let P 1,..., P n <x 1,..., x n >, a,j = ord xj P and = max σ S n Σ =1 n a,σ() r, then the order of P s not greater than.
Orgn of Jacob's bound Frst appeared n two posthumous papers: De nvestgando ordne systemats aequatonum dfferentalum vulgarum cujuscunque, Crelle s Journal 64, 1865, 297-320, De aequatonum dfferentalum systemate non normal ad formam normalem revocando, Vorlesungen über Dynamk 550-578, 1866. Edted by Cohn, Borchardt and Clebsch from a collecton of manuscrpts possbly wrten around 1840 (kept n the Archv der Berln-Brandenburgsche Akademe der Wssenschaft). Generc case
Rtt s contrbutons Jacob s bound n the lnear case, and for two varables. The dmenson conjecture for a sngle equaton. Paradoxcal examples: the ntersecton of two components of dfferental codmenson r 1 and r 2 may be of codmenson greater than r 1 + r 2 ; the ntersecton of two components of order e 1 and e 2 may be of codmenson greater than e 1 + e 2. The general case remans conjectural: let P 1,..., P n {x 1,..., x n }, be a prme component of {P} of dff. dm. 0, ord?
State of the art Jacob s weak bound for frst order systems. Lando 1970. Jacob s bound mples the dmenson conjecture. Cohn 1983. Jacob s bound for systems satsfyng Johnson s regularty hypothess. Kondrateva, Mkhalev and Pankratev 1982. Ordnary case. Extenson to the partal case. KMP 2008.
Provng the dmenson conjecture by extendng Rtt s method Let P1,..., P r {x 1,..., x n }, and be a prme component of {P}, we want to prove that the dfferental codmenson of s at most r. We make a change of varables: y = x η, where η s a generc zero of and get a new system Q. We complete the system wth n r generc lnear equatons and try to buld a Puseux seres soluton for Q: ζ = ζ 1 c + ζ 1 c α 1 + ζ 2 c α 2 + ζ 3 c α 3 +
Basc dea In Rtt s proof, each P s decomposed as a sum of homogeneous polynomals P = Σ k M k. Frst ζ 1 s chosen to be a generc soluton of M 1. If we have many equatons P = Σ k M,k, thngs reman unchanged provded that the M,1, completed wth the lnear equatons defne a component of dfferental dmenson 0. New terms n the Puseux seres are recursvely defned, so that the equatons could be satsfed up to a gven degree.
What must be changed Frst, the component may be of a smaller codmenson. Ths poses no dffculty for the proof, but we need to add more lnear relatons. The component of the deal {M,1 } may have a smaller codmenson. Then, e.g., M 1,1 (r) belongs to the radcal of the algebrac deal [M 1,1,M 1,1,..., M 1,1 (r) ] + {M,1 }. Some relaton between the Σ k>1 M,k must be used nstead of P 1.
The new polynomal s decomposed agan n homogeneous components, and we terate the process. As n Rtt s proof the soluton for the homogegeous system s njected n the orgnal system, and correctng terms must be ntroduced. Ths s here that the homogeneous sytem must have the rght codmenson. We have then a set of soluton dependng of at most n r arbtrary functons (due to the generc lnear equatons) and contanng η (takng c=0).
From the dmenson conjecture to the order bound We use Jacob s shortest reducton method. Let the l be the mnmal ntegers such that n the matrx a,j + l we may fnd n transversal maxma. We may compute characterstc sets (normal forms) for components of dff. Dm. 0 and order usng dervatves of P up to order l f Jacob s truncated determnant does not vansh. They are defned accordng to orderng on dervatve that are orderly wth respect to Jacob order defned by J. ord x = m.
Proof by recurrence on The case = 0 s straghtforward : the generators must be all of order 0. Let s assume the result s true for < s and consder a system P wth = s.
Geometrcal verson of Jacob s method. We start wth the equatons wth maxmal dervatves of smaller J. ord (those wth the maxmal l ). We take the radcal deal Q 0 they defne and for each prme component I n t... We take the radcal of the deal generated by the next set of equatons and I,..., I (l 1 -l 2 ) and for each prme component n t... We may terate the process unless the component contans elements of smaller J. ord not already n I,..., I (l 1 -l 2 ).
fle:///c:/users/ollver/desktop/101msdcf/dsc06336.jpg l 6 =0 l 5 l 4 l 3 l 2 l 1
The good case We have been able to terate the process up to the last set of generators wthout dscoverng new elements of non maxmal J. ord. The components of dff. Dm. 0 we have found are all of order.
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Another good case Some lower J. order element Q appeared... We may replace some P wth Q. The components of dff. Dm. 0 we look for wll be ncluded n those of ths new system... That has a smaller. Then, we use the recurrence hypothess.
A more complcated stuaton Sngular components : how to make them appear? Dfferentate more than the l. Very dffcult to control the order... Add extra equatons (separants). Usng some elmnaton, we may reduce to an extra equaton Q of smaller J. order than some P.
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Where we use the dmensonal property Consder the P j wth j. We may neglect the components they defne that are of codmenson smaller than ther number: they are ncluded n components of a system defned by droppng some equatons and of the same dmenson, so they cannot be used to make appear components of P of dff. dm. 0. Assume some dervatve of Q s n the deal generated by some of the remanng components I and lower dervatves of Q. Then t s enough to add only these lower dervatves to some regular component of {P}: the order must be less than.
The end If not, we may replace P, whch s greater than Q, by Q. The component we consder wll be n some component of the new system. And as Q has smaller J. order, the new system has smaller.