Differentia l Equations Standpoin t

America n Mathematica l Societ y

Colloquiu m Publication s Volum e 14

Differentia l Equation s from the Algebrai c Standpoin t

Josep h Fel s Ritt

America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/coll/014

2000 Mathematics Subject Classification. P r i m a r y 12-02 ; Secondar y 12H05 .

Library o f Congres s Cataloging-in-Publicat io n D a t a

Ritt, Josep h Fels , 1893-1951 . Differential equation s fro m th e algebrai c standpoint , b y Josep h Fel s Ritt .

p. cm . — (America n Mathematica l Societ y Colloquiu m publications , ISS N 0065-9258 ; v. 14 ) New York , America n mathematica l society , 1932 . ISBN 978-0-8218-4605- 6 (alk . paper ) 1. Differentia l equations . I . Title . II . Colloquiu m publication s (America n Mathematica l

Society) ; v. 14 .

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INTRODUCTION

We shal l be concerned , i n thi s monograph , wit h systems of differential equations , ordinar y o r partial, whic h ar e algebraic in th e unknown s an d thei r derivatives . Th e algebrai c sid e of th e theor y o f suc h system s seem s t o hav e remained , u p to th e present , i n a n undevelope d state .

It has been customary, in dealing with systems o f differential equations, t o assum e canonica l form s fo r th e systems . Suc h forms are inadequate for the representation o f genera l systems . It i s tru e tha t method s hav e been proposed for the reduction of genera l system s t o various canonica l types . Bu t the limit-ations whic h g o with the use of the implicit function theorem, the lac k o f method s fo r copin g wit h th e phenomen a o f degeneration whic h ar e eve r likel y t o occu r i n eliminatio n processes an d th e absenc e o f a techniqu e fo r preventing th e entrance o f extraneou s solutions , ar e merely symptoms of the futility inheren t i n suc h method s o f reduction .

Now, i n th e theor y o f system s o f algebrai c equations , on e witnesses a mor e enlivenin g spectacle . Kronecker' s Fest -schrift o f 188 2 se t upo n a firm foundatio n th e theor y o f algebraic eliminatio n an d th e genera l theor y o f algebrai c manifolds. Th e contribution s o f Mertens , Hilbert , KOnig , Lasker, Macaulay , Henzelt , Emm y Noether, va n der Waerden and others , hav e brought , t o thi s division o f algebra, a hig h degree o f perfection . I n th e notions o f irreducibl e manifold , and polynomia l ideal , ther e has been material for far reaching qualitative an d combinatoria l investigations . O n th e forma l side, on e ha s universall y vali d method s o f eliminatio n an d formulas fo r resultants .

To brin g t o th e theor y o f system s of differentia l equation s which ar e algebrai c i n th e unknown s an d thei r derivatives ,

iii i»

iv INTRODUCTION

some o f th e completenes s enjoye d b y th e theor y o f system s of algebrai c equations , i s th e ai m o f th e present monograph . The poin t o f vie w whic h w e tak e i s tha t o f ou r pape r Manifolds of functions defined by systems of algebraic differ-ential equations, published i n volum e 32 o f th e Transaction s of th e America n Mathematica l Society . I n wha t follows , we shal l outlin e ou r results .

Chapters I-VII I trea t ordinar y differentia l equations . W e deal wit h an y finite o r infinite syste m o f algebrai c differentia l equations in the unknown functions y u • • • , yn o f the variable a\ We writ e eac h equatio n i n th e for m

F{x\ yi,---,yn ) = 0 ,

where F i s a polynomia l i n th e yi an d an y number o f thei r derivatives. Th e coefficient s i n F wil l b e suppose d t o b e functions o f x, meromorphi c i n a give n ope n region . A n expression lik e F, above , wil l b e calle d a form. Al l form s considered i n thi s introductio n wil l b e understoo d t o hav e coefficients whic h ar e containe d i n a give n field. B y a field, we mea n a se t o f function s whic h i s close d wit h respec t t o rational operation s an d differentiation. *

Let 2 b e an y finite or infinite syste m o f forms i n yl9 • • • , y n* By a solution o f 2, w e mea n a solutio n o f th e syste m o f equations obtaine d b y settin g th e form s o f 2 equa l t o zero . The totalit y o f solution s o f 2 wil l b e calle d th e manifold of 2. I f 2 t an d 2% are system s suc h tha t ever y solutio n of 2 t i s a solutio n o f 2 2, w e shal l sa y tha t 2 2 holds 2 t.

A syste m 2 wil l b e calle d reducible or irreducible according as ther e d o o r d o no t exis t tw o forms , G and H, suc h tha t neither G no r H hold s 29 whil e GH hold s 2. Th e manifol d of 2, an d als o th e syste m o f equation s obtaine d by equatin g the form s o f 2 t o zero , wil l b e called reducible o r irreducible according a s 2 i s reducibl e o r irreducible .

We ca n no w stat e th e principal result o f Chapte r I . Every manifold is composed of a finite number of irreducible manifolds.

* A forma l definitio n i s give n i n § 1.

INTRODUCTION V

That is , give n an y syste m 2, ther e exis t a finite number of irreducible systems , 2 l9 •«• , 2S, suc h tha t 2 hold s ever y 2 t}

while ever y solutio n o f 2 i s a solutio n o f som e 2 t. Th e decomposition int o irreducibl e manifold s i s essentially unique.

Let u s conside r a n example . Th e equatio n

« ( & ) - * » - »•

whose solutions are y = (x — of, (a constant) , an d y = 0 , is a reducibl e syste m i n th e field of al l constants . Fo r

® £(U H holds th e first membe r o f (1) , whil e neithe r facto r i n (2 ) does. Th e syste m (1 ) i s equivalen t t o th e tw o irreducibl e systems

(g)'_4„ = 0, & - o and

The decompositio n theore m follow s fro m a lemm a whic h bears a certai n analog y to Hilbert's theorem on the existence of a finite basis fo r a n infinit e syste m o f polynomials . W e prove tha t if 2 is an infinite system of forms in y t, • • *, ynj

then 2 contains a finite subsystem whose manifold is identical with that of 2*

Chapters II an d VI stud y irreducibl e manifolds . W e start, in Chapte r II , wit h a precis e formulatio n o f th e notio n o f general solution o f a differentia l equation . W e d o no t thin k that suc h a formulatio n ha s bee n attempte d before . Le t A be a for m i n y Xf • • •, yn, effectivel y involvin g y n, an d irre -ducible, i n th e give n field, a s a polynomia l i n th e y% an d

* See § 124 fo r a comparison , wit h a theore m o f Tresse , o f th e ex-tension o f thi s lemm a to partia l differentia l equations .

vi INTRODUCTION

their derivatives . Le t th e orde r o f th e highes t derivativ e of y n i n A b e r an d le t y m represen t tha t derivative . Le t 2 b e th e totalit y o f form s whic h vanish fo r al l solution s o f A wit h dAldy nr # 0 . W e prov e tha t 2 i s irreducible . Th e manifold o f 2 i s on e o f th e irreducibl e manifold s i n th e decomposition o f th e manifol d o f A. W e cal l thi s manifol d the general solution of A (o r o f A = 0) .

The remainde r o f Chapte r I I deal s wit h th e association , with ever y irreducibl e syste m 2, o f a differentia l equatio n which w e cal l a resolvent o f 2. Th e first membe r o f th e resolvent i s a n irreducibl e polynomial , s o tha t th e resolven t has a genera l solution. Roughl y speaking , th e determination of th e genera l solutio n o f th e resolven t i s equivalent t o th e determination o f th e manifol d o f 2. Th e theor y o f resol -vents furnishe s a theoretica l metho d fo r th e constructio n o f all irreducibl e systems . On e will se e tha t th e resolven t ca n be use d advantageousl y i n forma l problems .

In Chapter VI, w e stud y what migh t be calle d th e textur e of a n irreducibl e manifold . Fo r th e cas e o f th e genera l solution of an algebraically irreducible form, our work amounts to characterizin g thos e singula r solution s (solution s wit h dA/dynr = 0) whic h belon g t o th e genera l solution .

Chapters V an d VI I contain , amon g othe r results , finite algorithms, involvin g differentiation s an d rationa l operations, for decomposin g a finite system int o irreducibl e systems an d for constructin g resolvents . I n Chapter V, w e d o no t obtai n the actua l irreducibl e systems , bu t rathe r certai n basic sets of form s (Ch . II) whic h characteriz e th e irreducible systems . However, thi s permit s th e constructio n o f resolvents . I n Chapter VII, a process is obtained which, if carried sufficientl y far, will actually produce the irreducible systems. Unfortunately , there i s nothin g i n thi s proces s whic h inform s one , a t an y point, as to whether or not the process has had its desired effect .

The result s o f Chapte r V furnis h a complet e eliminatio n theory fo r system s o f algebrai c differentia l equations .

In Chapte r VII , w e deriv e a n analogue , fo r differentia l forms, o f th e famou s Nullstellematz o f ffilbert and Netto. I n

INTKODUCTION vii

Chapter VIII , w e presen t a n analogu e o f Luroth' s theore m on th e parameterizatio n o f unicursai curves . I n Chapter III, there wil l b e foun d a theor y o f resultant s o f pair s of differ -ential forms. A number of other special results are distributed through th e monograph .

In Chapte r X, som e o f th e mai n result s state d abov e ar e extended to system s of algebraic partial differentia l equations . In particular , a n eliminatio n theor y i s obtaine d fo r suc h systems.

Chapter IV treats systems of algebraic equations. Th e chief purpose i s t o obtai n specia l theorems , an d finite algorithms, for applicatio n t o differentia l equatio n theory . Th e mai n results o f Chapte r I V ar e know n ones , bu t th e treatmen t appears new , an d som e specia l theorems , o f importanc e fo r us, d o no t see m t o exis t i n th e literature .

It ha s bee n ou r aim t o giv e this monograph an elementary character, an d t o assum e onl y suc h fact s o f algebr a an d analysis a s ar e containe d i n standar d treatises . Wit h thi s principle i n mind , w e hav e devote d Chapte r I X t o a n exposition o f Kiquier' s remarkabl e existenc e theore m fo r orthonomic system s o f partia l differentia l equations .

Thus Chapte r IX i s purel y expository , an d Chapte r IV i s largely so . Th e remaining chapters present result s containe d in ou r abov e mentione d paper , an d results communicate d b y us to the American Mathematical Society since the publication of tha t paper .

Koenigsberger's irreducibl e differentia l equations, * an d Drach's irreducibl e systems o f partia l differential equations, t are irreducibl e i n th e sens e describe d above . I n Drach' s definition, whic h include s tha t o f Koenigsberger , a syste m is calle d irreducibl e i f ever y equatio n whic h admit s on e solution o f th e syste m admit s al l solution s o f th e system . Thus, system s which ar e irreducibl e in ou r sens e ma y easil y be reducibl e i n th e theorie s o f Koenigsberge r an d Drach . The definition s o f Koenigsberge r an d Drach , whic h d o no t

* Lehrbuch der Differenzialgleichungen, Leipzig, 1889 . t Annate s d e FEcol e Normaie , vol . 34, (1898) .

viii INTRODUCTION

lead t o decomposition s int o irreducibl e systems , ar e th e starting points of group-theoretic investigations, which parallel the Galoi s theory . Ou r course , a s w e hav e seen , i s i n a different direction .

Many questions still remain for investigation. I n particular , a theor y o f ideal s o f differentia l form s an d a theor y o f birational transformations, awai t development.* Chapter s VII and VIII ma y perhaps be regarded as rudimentar y beginnings of suc h theories .

It goe s withou t sayin g tha t w e hav e bee n guided , i n ou r work, b y th e existin g theor y o f algebrai c manifolds . W e have foun d particularl y valuable , th e excellen t treatmen t o f systems o f algebrai c equation s give n i n Professo r va n de r Waerden's pape r Zur Nullstellentheorie der Polynomideale.i But it is not surprising, on the other hand, that the investigation of essentiall y ne w phenomen a shoul d hav e calle d fo r th e development o f ne w methods .

I a m ver y gratefu l t o th e Colloquiu m Committe e o f th e American Mathematica l Society , wh o hav e invite d m e t o lecture o n th e subjec t o f thi s monograp h a t th e Universit y of California in September, 1932. T o my friend an d colleagu e Dr. Eli Gourin, who assisted me in reading the proofs, I extend my dee p thanks .

*In connectio n wit h transformation s o f genera l (non-algebraic ) differ -ential equations, se e Hilbert, Mathematisch e Annalen, vol . 73 (1913) , p . 95.

t Mathematisch e Annalen , vol . 96, (1927) , p . 183.

NEW YOKK , N . Y .

February, 1932 . J . F . KlTT .

CONTENTS

Page CHAPTER I

DECOMPOSITION O F A SYSTE M O F ORDINAR Y ALGEBRAI C

DIFFERENTIAL EQUATION S INT O IRREDUCIBL E SYSTEM S . 1

Fields, forms , ascendin g sets , basi c sets , reduction , solution s and manifolds , completenes s of infinite systems , non-existenc e o f a Hilbert theorem , irreducibl e systems , th e fundamental theorem , uniqueness o f decomposition , examples , relativ e reducibility, ad -junction o f ne w unknowns , fields o f constants .

CHAPTER I I GENERAL SOLUTION S AN D RESOLVENT S 2 1

General solutio n o f a differentia l equation , close d systems , arbitrary unknowns , th e resolvent , invarianc e o f th e intege r g , order o f th e resolvent , constructio n o f irreducibl e systems , irre -ducibility an d th e ope n regio n ST .

CHAPTER H I FIRST APPLICATION S O F TH E GENERA L THEOR Y 4 7

Resultants o f differentia l forms , analogu e o f an algebraic theo-rem o f Kronecker, for m quotients .

CHAPTER I V SYSTEMS O F ALGEBRAI C EQUATION S 6 2

Indecomposable system s o f simpl e forms , simpl e resolvents , basic sets o f prim e systems, construction o f resolvents, resolutio n of a finite system into indecomposable systems, a special theorem.

CHAPTER V CONSTRUCTIVE METHOD S 9 2

Characterization o f basic sets of irreducibl e systems , basi c sets in a resolution o f a finite system into irreducible systems, tes t fo r a form to hold a finite system, construction of resolvents, a remark on th e fundamenta l theorem , Jacobi-Weierstras s canonica l form.

ix

x CONTENT S

Page CHAPTER V I

CONSTITUTION O F A N IRREDUCIBL E MANIFOL D 10 0

Seminormal solutions, adjunctio n o f new functions to §r , inde-composability an d irreducibility .

CHAPTER V H ANALOGUE O F TH E HILRERT-NETT O THEOREM , THEORETICA L

DECOMPOSITION PROCES S 10 8

Analogue o f Hilbert-Nett o theorem ; theoretica l proces s fo r decomposing a finite sjstem o f form s int o irreducibl e systems ; forms i n on e unknown , o f first order .

CHAPTER VH I ANALOGUE FO R FOR M QUOTIENT S O F LUROTH' S THEORE M 12 4

CHAPTER I X RIQUIER'S EXISTENC E THEORE M FO R ORTHONOMI C SYSTEM S 13 5

Monomials, dissectio n o f a Taylo r series , marks , orthonomi c systems, passiv e orthonomi c systems .

CHAPTER X SYSTEMS O F ALGERRAI C PARTIA L DIFFERENTIA L EQUATION S 15 7

Decomposition of a system into irreducible systems, basic sets of close d irreducibl e systems , algorith m fo r decomposition , analogue o f th e Hilbert-Nett o theorem .

INDEX The nnmbers refer t o section s

Adjunction o f unknown s 1 7 arbitrary unknown s 2 4 ascending se t 3 Basic se t 4

Class o f form s 2 complete se t 10 2 Extended se t 10 2 Field 1 fields of constant s 1 8 form 2,12 0

— algebraicall y irreducibl e . 1 9 — genera l 3 4 — quotient s 3 8 — simpl e 4 1

Hilbert-Netto theore m 77,12 9 Indeterminate 3 4 initial 5,12 3 integrability condition s 11 8 Jacobi-Weierstrass for m 7 1 Kronecker's theore m 3 5 Leader 12 1 Liiroth's theore m 9 1 Manifold 6 mark 10 6 monomial 9 9

multiple 9 9 multiplier 10 3

Open region 1 Parametric derivativ e 10 8 principal derivativ e 10 8 Rank 2,12 1 relative reducibilit y 1 6 remainder 5,12 3 resolvent 2 9 — simpl e 4 2

resultant 3 4 Separant 5,12 3 solution 6 — genera l 1 9 — normal 7 2 — regular 20,23,12 5 — seminorma l 7 2

system — close d 2 3 — complet e 7 — essentia l irreducibl e 1 4 — indecomposable 41 — irreducible 1 2 — orthonomi c 10 8 — non-trivia l 2 3 — passiv e 11 9 — prime 4 2

— simpl y close d 4 1 systems equivalen t 1 3

172