Commented bibliographical references
1 Two posthumous papers    2 Manuscripts    3 Later works    4 Affectation problem    5 Jacobi's biography    6 Translations from latin
                                                                        3.1 The xixth century

                                                                                                3.2 The beginning of the xxth century
                                                                                                3.3 The end of the xxth century
                                                                                                3.4 xxith siècle

 

1  Two posthumous papers

The main sources are the two papers [20, 21]. The first introduces the bound and gives a sketch of a proof. It then describes an algorithm allowing to solve the affectation problem. After having recalled the bound and the algorithmm, the second shows how one may compute a normal form using as few derivatives of each equation as possible and then how to eliminate all variables except one, using again as few derivatives of each equation as possible.

2  Manuscrits

A letter from Sigismund Cohn [II/13 a)], who initiated the editing work, pursued after his death by Borchardt, gives us details about the manuscripts left by Jacobi.

He used parts of various fragmentary versions, gathered in [II/13 b)] in order to make a text as coherent as possible, that was later modified by Borchardt [II/13 c)]. The manuscript [I/63] is the final version intended to the typographer.

The paper [21] has been derived from the manuscript [II/23 b)]. There should have been a beginning of a transcription by Cohn, later pursued by Borchardt. One finds in [II/25] an abstract from Borchardt's hand and an index with references to pages in both transcription. There is also a part in latin, from Borchardt's hand, that may be a sketch for a transition text not retained in the final version.

The manuscript [II/24], from an unknown hand, mixes the texts of both papers. The latin is of a lower quality, but the writing is fine. A copist work for a synthesis of both texts?

The manuscripts [II/22, II/23 a)] consider the problem of computing a normal form in view of the determination of multipliers. That is in the second part of the paper [23], initialement published in [Crelle29] that jacobi mentioned for the first time his method foor the computation of a normal form. The manuscript [I/58 a)], dated from Roma December 27th, 1843 corresponds to the first part of this paper, initially published in [Crelle27].

3  Later works

3.1  The XIXth century

Edward John Nanson (1850-1936) [46] and Camille Jordan [30] have claimed to bring more rigorous proofs, the arguments of which are only generically true. George Chrystal [4] considered the linear case with constant coefficients. the famous paper of Sofya Kovalevskaya [38] is the only one for the best of my knowledge, with that of Ritt [53], to quote [21].

3.2  First half of the XXth century. Ritt

Joseph fels Ritt proved the bound in the linear case in [53] and the non linear case for two variables in [54].

3.3  Second half of the XXth century

The paper [57] of Leonid Romanovich Volevich contains a proof of the bound, rediscovered independently of Jacobi's work, for a non degenerated linear system, I.e. a system such that the truncated determinant is non vanishing.

In the framework of differential algebra, the main works in this period are those of Richard M. Cohn's students. Bernard Greenspan [15, 16] proved for a system defining components being all of dimension 0 a bound that is weaker than Jacobi's but better than Bézout's analog in the case of difference equations. Barbara Lando [39, 40] proved the weak bound1 for order 1 difference system, a result generalized to differential systems [41]. Richard Cohn a proved Greenspan's bound for arbitrary differential systems [5] and shown that Jacobi's bound implies the dimensional conjecture2 [6].

Marina Vladimirovna Kondratieva et al. [32, 33] have proved the bound under some regularity hypotheses allowing to reduce to the linear case. Under similar hypotheses, Joseph Johnson proved Maurice Janet's conjecture [29]. About linearizing differential systems and Kaehler's differential in differential algebra, see [27, 28].

Let us mention an attempt to generalize the bound to partial differential systems, due to Joseph S. Tomasovic, Jr. [56], a student of Ellis Robert Kolchin.

Outside the circle of differential algebar, Jacobi's bound is not very well known. We may signal the works of Robert J. Magnus in the case of linear differential operators that refer to Jacobi and Chrystal [42, 43].

3.4  The XXIth century

Ehud Hrushovski gave in [19] a proof of the bound for difference systems.

4  The affectation problem

The excellent paper of Alexander Schrijver [55] gives a quite complete historical overview, to the exception of Jacobi's work, unknow to that scientific community when the paper has been written. The problem considered by Monge in [44] may be considered as a special case.

Dénes Kőnig gave a pollynomial time algorithm for searching a maximal matching in a bipartite graph, that is equivalent to consieder a matrix of 0 and 1 [34].

See [18] for a faster method. See also [7] for more details.

A footnote in [35] indicated a generalization to the case of weighted graphs by Jeno Egerváry [8]. This result allowed H. Kuhn to develop is de développer sa “Hungarian method” [36, 37, 45, 10].

See also [9].

5  Jacobi's biography

The book [31] is a useful reference, despite his hagiographical style. Hebert Pieper gives in [49] an historical study and precious indications on sources locations. The very complete list [1] provides all the publication by and about Jacobi available at the library of the science academy in Berlin.

6  Other translations of Jacobi's texts in latin

The paper of N. Gauthier [12] contains a translation of a short passage of [26] that gives an original solution to the problem of the motion of a point subject to a gravitational force.

The memoire of Laure Buhry [2] on the sur le theorem AF + BG of Noether contains a full translation of [25].

References

6.1  Manuscrits

[I/58 a)]
Theoria novi multiplicatoris systemati aequationum differentialium vulgarium applicandi, arbeit von Schreibers Hand, Ergänzungen und Korrekturen von Jacobi und Crelle. 1843. 71  S. (unvollständig) [Correspond aux deux premiers chapitres de [23], p. 317–394 de [GW4] et [C27 p. 199–268 Heft III]. La page 1 du manuscrit est datée “Roma d. 27 Dec. 1843”. Cursives latines. Indications typographiques marginales en Kurrentschrift (p. 1, 5...).]


[I/63]
De investigando ordine systematis aequationum differentialum vulgarium cujuscunque. (Nach dem Tode Jacobis von C.W. Borchardt herausgegeben.) Manuscript von Bochardts Hand; Fragment. 20 S. [P. 9–28 (soit les 20 dernières pages) du manuscrit de [20] destiné au typographe. En cursives latines pour le texte, en Kurrentschrift pour les indications typographiques marginales.]


[I/67a]
Vorlesungen über Dynamik. von C.G.J. Jacobi. Berlin 1866. Titelblatt, Vorwort und Inhaltsverzeichnis. (Durchschoss. Druck mit Korrekturen von fremder Hand. Nicht in G.W. abgedruckt.)

anschließend:

[I/67b]
De aequationum differentialum systemate non normali ad formam normalem revocando. Druck aus dem von Clebsch herausgegeben Vorlesungen über Dynamik von C.G.J. Jacobi Berlin 1866. S. 550–578, mit Korrekturen von Weierstraßund vermutl. Borchardt. [cf[21]]


[II/10]
1 Umschlag, enthaltend: Papiere zu meiner [Borschardts] und Cohns Abschrift von Jacobis Abhandlungen über Reduktion simultaner Differentialgleichungen in die canonische Form gehörtig, die aber ungeeignet sind, an Clebsch gesandt zu werden


[II/11 b)]
Fragment. 37 S. Von Jacobis Hand. Auf Beiblatt zu S.15 ist die Jareszahl 1848 vermerkt.


[II/13 a)]
Cohn: Schreiben, vermutlich an C.W. Borchardt. Hirschberg, 25.8.1859. 3 S.

[II/13 b)]
Jacobisches Manuscript: De ordine systematis aequationum differentialium canonici variisque formis quas inducre potest. Rote Zahl: 2186–96, 2200–2206. 35 S. Grundlage der von Cohn abgeschriebene Abhandlung.

[II/13 c)]
Cohn [Sigismund]; Abschrift der Bll. 2205, 2206, 2204, 2203, 2202, 2201, 2200, 2187, 2188, 2189, 2196, 2195, 2191, 2192, 2193, 2194, Mit einem von Borchardt versehenen Inhaltsregister. 39 S.


[II/22]
Jacobi: De reductione simplicissima systematis aequationum differentialium ad formam canonicam. Rote Zahl: 2182–2213. Darin enthalten die Seiten 2182–2185, 2197–2199, 2207–2213. Das Manuskript ist von Jacobis Hand. 14 Bll.


[II/23]
De multiplicatore systematis aequationum differentialium forma normali non gaudentes. Rote Zahl: 2214–2251. Drin enthalten:

[II/23 a)]
Reduction simultaner Differentialgleichungen in ihre canonische Form und Multiplicator derselben. Rote Zahl: 2214–2237. Manuskript von Jacobi. 43 S. Umschlagvermerk: “Wahrscheinlich unbrauchbar, wenigs vorläufig” von Borchardt.

[II/23 b)]
Titel wie auf dem Umsclag bzw. De aequationum differentialium systemate non normali ad formam normalem revocando. Bll. 2238, 2239–2241, 2242–2251. 25 S. Ms. von Jacobi, Umschlag von Borchardt.


[II/24]
De aequationum differentialium systemate non normali ad formam normalem revocando.

Anschließend : De investigando ordine systematis aequationum differentialum vulgarium cujuscunque.

[Fragment] von fremder Hand. 11 S.


[II/25]
De aequationum differentialium systemate non normali ad formam normalem revocando. Notizen von Borchardts Hand. 8 S.


[V/6]
Cohn. S[igismund]: Ueber confocale Flächen 2ten grades.

[V/6 a1)]
Manuscript von Cohn Entwurf. 24 S. mit 2 Abb.

[V/6 a2)]
Beginn der Reinschrift von Cohns Hand. 2 S. mit 2 Skizzen. Mit Vermerk von Freyer.

[V/6 b)]
Abschrift des Manuskriptes von Freyers Hand. Datiert: Schweidnitz, März 1864, nach dem Tode Cohns. 14 S. ohne Abb.

6.2  OEuvres complètes



[FD]
Vorlesungen über Dynamik von C. G. J. Jacobi nebstes fünf hinterlassenen Abhandlungen desselben, herausegegeben von A. Clesch, Berlin, Druck und Verlag von Georg Reimer, 1866.

[GWIV]
C.G.J. Jacobi's gesammelte Werke, vierter Band, herausgegeben von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1890.

[GWV]
C.G.J. Jacobi's gesammelte Werke, fünfter Band, herausgegeben von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1890.

6.3  Journal de Crelle



[Crelle27]
Crelle Journal für die reine und angewandte mathematik, Bd. 27, 1844.

[Crelle29]
Crelle Journal für die reine und angewandte mathematik, Bd. 29, 1845.

[Crelle64]
Borchardt Journal für die reine und angewandte Mathematik, Bd LXIV, Heft 4, p. 297-320, Berlin, Druck und Verlag von Georg Reimer, 1865

6.4  Publications



[1]
Karl Gustav Jacob Jacobi, Mathematiker, Ausgewählte literaturnachweise aus dem Bestand der Akademiebibliothek, Berlin-Brandenburgische Akademie der Wissenschaften, Akademiebibliothek, 2002.

[2]
Buhry (Laure), Le théorème AF + BG de Noether, Mémoire de Master recherche, université Bordeaux 1, 2006.

[3]
Buium (Alexandru), Differential algebra and diophantine geometry, Actualités Mathématiques, Hermann, Paris, 1994.

[4]
Chrystal, Transactions of the Royal Society of Edinburgh, vol. 38, p. 163, 1895.

[5]
Cohn (Richard M.), “The Greenspan bound for the order of differential systems”, Proc. Amer. Math. Soc. 79 (1980), n∘ 4, 523–526.

[6]
Cohn (Richard M.), “Order and dimension”, Proc. Amer. Math. Soc. 87 (1983), n∘ 1, 1–6.

[7]
Cormen (Thomas H.), Leierson (Charles E.), Rivest (Ronald L.) et Stein (Clifford), Introduction to algorithms, second edition, The MIT Press, Cambridge, 2001.

[8]
Egerváry (Jenő) [Eugène], “Matrixok kombinatorius tulajdonságairól” [En Hongrois: Sur les propriétés combinatoires des matrices], Matematikai és Fizikai Lapok, vol. 38, 1931, 16–28; traduit en américain par H. W. Kuhn as Paper 4, Issue 11 of Logistik Papers, Georges Washington University Research Project, 1955.

[9]
Ford, Jr. (L.R.) et Fulkerson (D.R.), “Solving the transportation problem”, Management Sci., 3, 24–32.

[10]
Franck (András), On Kuhn's Hungarian method — A tribute from Hungary, Egerváry Research Group on Combinatorial Optimization, Technical reports, TR-2004-14, Budapest, 2004, 7 p. ISSN 1587-4451.

[11]
Gaffiot (Félix), Dictionnaire illustré Latin Français, Librairie Hachette, Paris, 1934. Réédition complétée, 1978, ISBN 2.01.000535.X

[12]
Gauthier (N.), “Jacobi's 1842 solution of the inverse-square problem”, AM. J. Phys. 72, Mars 2004.

[13]
Gourin (E.), “??? ”, Bulletin of the Am. Math. Soc, vol. 39 (1933), p. 593.

[14]
Grévisse (Maurice), Le bon usage. Grammaire française, refondue par André Goose, treizième édition revue, Duculot, Paris, Louvain-la-Neuve, 1993.

[15]
Greenspan (Bernard), A bound on the order of the Zero dimensional components of a system of algebraic difference equations, Ph.D. dissertation, Rutgers University, New Brunswick, 1959.

[16]
Greenspan (Bernard), “A bound for the orders of components of a system of algebraic difference equations”, Pacific J. Math., 9, 1959, 473–486.

[17]
Hoe (John), Les systèmes d'équations polynômes dans le siyuan yujian (1303), Mémoires de l'institut des hautes études chinoises, vol. VI, Collège de France, Institut des hautes études chinoises, [Paris 1977 ?]. ISBN 2-85757-004-X.

[18]
Hopcroft (John E.) et Karp (Richard M.), “An n5/2 algorithm for maximum matchings in bipartite graphs”, SIAM Journal on Computing, 2 (4), 225–231, 1973.

[19]
Hrushovski (Ehud), The Elementary Theory of the Frobenius Automorphisms, preprint,http://arXiv.org/abs/math/0406514, 2004.

[20]
Jacobi (Carl Gustav Jacob),“De investigando ordine systematis aequationum differentialum vulgarium cujuscunque ”, publié par C. W. Borchardt, Borchardt Journal für die reine und angewandte Mathematik, Bd LXIV, Heft 4, p. 297-320, Berlin, Druck und Verlag von Georg Reimer, 1865, reproduit dans C.G.J. Jacobi's gesammelte Werke, fünfter Band, herausgegeben von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1890, p. 193-216. (cf[I/63])

[21]
Jacobi (Carl Gustav Jacob), “De aequationum differentialum systemate non normali ad formam normalem revocando”, Vorlesungen über Dynamik von C. G. J. Jacobi nebstes fünf hinterlassenen Abhandlungen desselben, herausegegeben von A. Clesch, Berlin, Druck und Verlag von Georg Reimer, 1866, p. 550–578 et C.G.J. Jacobi's gesammelte Werke, fünfter Band, herausgegeben von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1890, p. 485-513. (cf[I/67 a) et b)])

[22]
Jacobi (Carl Gustav Jacob), “Theoria novi multiplicatoris systemati aequationum differentialium vulgarium applicandi”, Crelle Journal für die reine und angewandte Mathematik, Bd. 27 Heft III, p. 199–268 [chap. I et II] (1844), Bd. 29 Heft III p. 213–279 [chap. III sections 14 à 25], Heft IV 333–376 [chap. III sections 26 à 32] (1845) [Daté “Berol. d. 26 Julii 1845”], C.G.J. Jacobi's gesammelte Werke, vierter Band, herausgegeben von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1886, p. 495–509. (cf[I/58a])

[23]
Jacobi (Carl Gustav Jacob), “Sul principo dell'ultimo moltiplicatore e suo uso come nuovo principo generale di meccanica”, Giornale arcadico, Tomo XCIX, p. 129–146, C.G.J. Jacobi's gesammelte Werke, vierter Band, herausgegeben von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1886, p. 513–522.

[24]
Jacobi (Carl Gustav Jacob), “Geometrische Theoreme”, [Fragments publiés par O. Hermes], Crelle Journal für die reine und angewandte mathematik, Bd. 73, p. 179–206.

[25]
Jacobi (Carl Gustav Jacob), “De relationibus, quae locum habere debent inter puncta intersectionis duarum curvarum vel trium superficierum algebraicarum dati ordinis, simul cum enodatione paradoxi algebraici”, Crelle Journal für die reine und angewandte Mathematik, Bd. 15, 1836, p. 285–308, reproduit dans C.G.J. Jacobi's gesammelte Werke, dritter Band hrsg. von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1884, 329–354.

[26]
Jacobi (Carl Gustav Jacob), “De motu puncti singularis”, C.G.J. Jacobi's gesammelte Werke, vierter Band, herausgegeben von K. Weierstrass, Berlin, Druck und Verlag von Georg Reimer, 1886, p. 495–509.

[27]
Johnson (Joseph), “Differential dimension polynomials and a fundamental theorem on differential modules”, Amer. J. Math. 91, 239–248, 1969.

[28]
Johnson (Joseph), “Kähler differentials and differential algebra”, Ann. of Math. 89, 92–98, 1969.

[29]
Johnson (Joseph), “Systems of n partial differential equations in n unknown functions: the conjecture of M. Janet”, Trans. of the AMS, vol. 242, Aug. 1978.

[30]
Jordan (Camille), “Sur l'ordre d'un système d'équations différentielles ”, Annales de la société scientifique de Bruxelles, vol. 7, B., 127–130, 1883.

[31]
Koenigsberger (Leo), Carl Gustav Jacob Jacobi. Festschrift zur Feier der hundertsten Wiederkehr seines Geburstages, B. G. Teubner, Leipzig, 1904, xviii, 554 p.

[32]
Кондратьева (Марина Владимровна),  «О  границе Якоби системы обыкновенных дифференциальных уравнений», Y  Всесоюзный  симпозиум по  теории  колец, алгебр и  модулей (21-23 сентября 1982г.): Тезисы сооб. Новосибирск: Институт математики СО АН СССР, 1982, с. 74.

[33]
Кондратьева (Марина Владимровна),  (Михалев Алеандр Васильевич), Панкратьев (Евгени Васильевич), «О  границе  Якоби  для систем обыкновенных дифференциальных  многочленов», Алгебра.  М.: МГУ, 1982, с. 79-85.

[34]
Konig (Dénes), “Graphok és matrixok”, Matematikai és Fizikai Lapok, 38 (1931), 116–119.

[35]
Konig (Dénes), Theorie der endlichen und unendlichen Graphen, (1936), Chelsey, New-York, 1950.

[36]
Kuhn (Harold H.), “The Hungarian method for the assignment problem”, Naval res. Logist. Quart. 2 (1955), 83–97.

[37]
Kuhn (Harold H.), “Variants of the Hungarian method for assignment problems”, Naval res. Logist. Quart. 3 (1956), 253–258.

[38]
Kowalevsky (Sophie von) [Sofya Kovalevskaya], “Zur Theorie der partiellen Differentialgleichungen”, Journal für die reine ound angewandte Mathematik, 80, 1875, 1–32.

[39]
Lando (Barbara A.), Jacobi's bound for the order of systems of first order differential equations, Ph.D. dissertation, Rutgers University, New Brunswick, 1969.

[40]
Lando (Barbara A.), “Jacobi's bound for the order of systems of first order differential equations”, Trans. Amer. Math. Soc. 152 1970, 119–135.

[41]
Lando (Barbara A.), “Jacobi's bound for first order difference equations”, Proc. Amer. Math. Soc. 32 1972, 8–12.

[42]
Magnus (Robert J.), “Operator-valued functions, multiplicity and systems of linear differential equations”, skýrsla RH-20-2001, Raunvísindastofnun Háskólans, 2001.

[43]
Magnus (Robert J.), “Línuleg diffurjöfnuhneppi og setningar Jacobis og Chrystals”, Tímarit um raunvísindi og stærðfræði, 1. árg. 2 hefti, 2003.

[44]
Monge (Gaspard), “Mémoire sur la théorie des déblais et des remblais”, Histoire de l'Académie royale des Sciences, [année 1781. Avec les Mémoires de Mathématique & de Physique pour la même Année] (2e partie) (1784) [Histoire: 34­38, Mémoire :] 666­704.

[45]
Munkres (James), “Algorithms for the assignment and transportation problems”, J. Soc. Industr. Appl. Math., 5 (1957), 32–38.

[46]
Nanson (E.J.), “On the number of arbitrary constants in the complete solution of ordinary simultaneous differential equations”, Messenger of mathematics (2), vol. 6, 77–81, 1876.

[47]
Ollivier (François) et Sadik (Brahim), La borne de Jacobi pour une diffiété définie par un système régulier, prépublication, 2006.

[48]
Pascal (Blaise), OEuvres complètes, Bibliothèque de la Pléiade n∘ 34, Gallimard, Paris, 1954.

[49]
Correspondenz Adrien-Marie Legendre – Carl Gustav Jacob Jacobi, éditée par Herbert Pieper, avec une étude “C. G. J. Jacobi in Berlin”, B. G. Teubner Stuttgart, Leipzig, 1998, 245 p.

[50]
Ritt, (Joseph Fels), “On the differentiability of a differential equation with respect to a parameter”, Ann. of Math. (2) 20 (1919), n∘ 4, 289–291.

[51]
Ritt, (Joseph Fels), Differential Equations from the Algebraic Standpoint, Amer. Math. Soc. Colloq. Publ., vol. 14, A.M.S., New-York, 1932.

[52]
Ritt, (Joseph Fels), “Systems of algebraic differential equations”, Annals of Mathematics, vol. 36, 1935, 293–302.

[53]
Ritt, (Joseph Fels), “Jacobi's problem on the order of a system of differential equations ”, Annals of Mathematics, vol. 36, 1935, 303–312.

[54]
Ritt, (Joseph Fels), 1950. Differential Algebra, Amer. Math. Soc. Colloq. Publ., vol. 33, A.M.S., New-York.

[55]
Schrijver (Alexander), “On the history of combinatorial optimization (till 1960)”, Handbook of Discrete Optimization, K. Aardal, G.L. Nemhauser, R. Weismantel, eds., Elsevier, Amsterdam, 2005, pp. 1–68.

[56]
Tomasovic, Jr., (Joseph S.), A generalized Jacobi conjecture for arbitrary systems of differential equations, Dissertation, Columbia University, 1976. [52 feuillets, 29 cm. Columbia University Rare Books and Manuscript Library, Butler Library, New-York, LD1237.5D 1976 T552]

[57]
Volevich (Leonid Romanovich), <Ob obshchikh sistemax differentsial'nykh uravnenui>, Doklady AN SSSR, 1960, t. 132, N0 1, 20–23. Traduction anglaise: “On general systems of differential equations”, Soviet. Math. 1, 1960, 458–465.
1
With the convention ordxjPi=0 if Pi does not contain xj and its derivatives. With the convention ordxjPi=−∞ in such a case it is the strong bound. Proving the strong bound for order 1 systems imply the general case (see [56]).
2
The components defined by r equations have codimension at most r.
This document was translated from LATEX by HEVEA.