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Jacobi’s Bound: Jacobi’s results translated in Kőnig’s, Egerváry’s and Ritt’s mathematical languages

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Abstract

Jacobi’s results on the computation of the order and of the normal forms of a differential system are translated in the formalism of differential algebra. In the quasi-regular case, we give complete proofs according to Jacobi’s arguments. The main result is Jacobi’s bound, still conjectural in the general case: the order of a differential system \(P_{1}, \ldots , P_{n}\) is not greater than the maximum \({{\mathcal {O}}}\) of the sums \(\sum _{i=1}^{n} a_{i,\sigma (i)}\), for all permutations \(\sigma \) of the indices, where \(a_{i,j}:=\mathrm{ord}_{x_{j}}P_{i}\), viz. the tropical determinant of the matrix \((a_{i,j})\). The order is precisely equal to \({{\mathcal {O}}}\) iff Jacobi’s truncated determinant does not vanish. Jacobi also gave a polynomial time algorithm to compute \({{\mathcal {O}}}\), similar to Kuhn’s “Hungarian method” and some variants of shortest path algorithms, related to the computation of integers \(\ell _{i}\) such that a normal form may be obtained, in the generic case, by differentiating \(\ell _{i}\) times equation \(P_{i}\). Fundamental results about changes of orderings and the various normal forms a system may have, including differential resolvents, are also provided.

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Notes

  1. References to manuscripts and first editions are given page 92.

  2. Jacobi himself had an experience in practical computing, on a smaller scale and in a different field, when he published his Canon arithmeticus [47]. The revision of the half million numbers it contains required the help of friends and relatives [8], including Dirichlet’s wife and mother!

  3. From some optimistic standpoint, it could have been a way to escape ethical issues raised by the use of psychology in management.

  4. The existence of a canon is a consequence of Algorithm 9, unicity is shown in Proposition 7 below.

  5. Jacobi defined also the maximal elements in right columns as “starred”; we prefer to reserve this denomination to left transversal maxima to underline the specific roles played by these two sets of maxima in the algorithm.

  6. This notion is closely related to that of increasing path, as defined in Hopcroft and Karp [34] (see 3.1), which explains the choice of that word to translate transitum datur in [40].

  7. Determinans mancum, or Determinans mutilatum.

  8. Such a decomposition is known to be unique, see Ritt [86, ch. II Sect. 19] or Kolchin [59, th. 1, p. 14].

  9. Ritt [86, chap. II Sect. 17 p. 32] defines singular components only in the case of an ideal generated by a single polynomial.

  10. The meaning of this classical notion of regularity is of a different nature and will be investigated in Sect. 7.4

  11. We refer to Castro-Jiménez [10, 11] for more details on standard (or Gröbner) bases of \({{\mathcal {D}}}\)-modules.

  12. Jacobi used many different notations for his bound. In our translation [40], we used H for consistency, following Borchardt’s posthumous edition [Crelle 64]. The bound is actually denoted by \(\mu \) in the manuscript [II/13 b)]. See [40, fig. p. 32]. We prefer here the notation \({{\mathcal {O}}}\), used in [II/23 b)], as H is now a standard notation in differential algebra for the product of initials and separants.

  13. We allow ourselves to write instead of \(a_{{{\mathcal {P}}}\!,\,i,\,j}\) for more readability.

  14. In the linear case, the result has been proved by Ritt [85].

  15. Looking for the order of the system, as one only considers the highest derivatives in the linear equations to which the proposed ones are reduced, one may assume [their] coefficients to be constants. Because, differentiating the equations 3) iterated times in order to obtain new equations [...]

  16. Cf. Castro-Jiménez [10, 11].

  17. Another notion of “regularity” appears here, viz. the possibility to test ideal membership by pseudo-reduction, using its characteristic set. But we will see in Sect. 7.4 that the corresponding “regular components” are also regular according to Definition 84 iii).

  18. See Houtain [35] and the comments of Ritt on the approaches of Lagrange [86, p. 33] or Laplace, Poisson and Hamburger [86, p. 77]

  19. The orders \(a_{i}\) or \(b_{i}\) correspond to \(\alpha _{i}\) or \(\beta _{i}\) in Jacobi’s notations, in order to reserve these Greek letters to covers. This writing can coexist with the order matrix notation \(a_{i,j}=\mathrm {ord}_{x_{j}}A_{i}\), so that \(a_{i}\) is a convenient shorthand for \(a_{i,i}\).

  20. These orders are already in Riquier [83, p. 195].

  21. As we are only concerned here with the differential case, we omit “differential” in the sequel.

  22. The word resolvent was not used by Jacobi, but he evokes the notion as something well known in the the mathematical folklore of his time: “It is usual that this type of normal forms be considered before others by mathematicians”.

  23. Nanson [76] and Jordan [54] proposed independently heuristic methods for proving Jacobi’s bound, that rely on resolvent computations. The first considers the case \(n=3\) and the second the case \(n=4\), recursively using formula \({{\mathcal {O}}}=\max _{i} a_{i,j_{0}}+{\bar{{{\mathcal {O}}}}}_{i,j_{0}}\) and the bound \({\bar{{{\mathcal {O}}}}}_{i,j_{0}}\) on the order of differentiation of each equation \(P_{i}\), which is guessed using informal considerations on the number of derivatives of the equation \(P_{i}\), and the number of derivatives of the \(x_{j}\), \(j\ne j_{0}\) that the computation of a resolvent requests. Their relation is expressed by the formula

    $$\begin{aligned} \sum _{i=1}^{n}\left( {\bar{{{\mathcal {O}}}}}_{i,j_{0}}+1\right) =1+\sum _{j\ne j_{0}}\left( \max _{i=1}^{n}({\bar{{{\mathcal {O}}}}}_{i,j_{0}}+a_{i,j})+1\right) , \end{aligned}$$

    so that there are exactly one more algebraic equations than derivatives of the \(x_{j}\), \(j\ne j_{0}\), involved in them, making their potential elimination possible. The last formula is proved in Corollary 173.

  24. It may be computed as in Sect. 3.6. By Proposition 54, the path relation does not depend on the choice of the permutation \(\sigma \). The canon itself is also independent of this choice: if \(a_{j_{0},j_{0}}+\ell _{j_{0}}\) is maximal, then for any permutation \(\sigma \) such that \({{\mathcal {O}}}=\sum _{i=1}^{n}a_{i,\sigma (i)}\), the quantity \(a_{\sigma ^{-1}(j_{0}),j_{0}}+\ell _{\sigma ^{-1}(j_{0})}\) must be maximal too.

  25. One may look at formula (3) p. 14 for an illustration of the situation. The notion of increasing path is used here in the reverse way: one deduces a maximal transversal sum for \({{\bar{A}}}_{i_{0},j_{0}}\) from a maximal transversal sum for \(A_{P}\).

  26. Reference established from Bull. Amer. Math. Soc. 12 (1906), p. 212. In the copy I could consult, there is no publisher indication, and the handwritten date 1852.

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Acknowledgements

With the support of Agence Nationale de la Recherche, project ANR-10-BLAN 0109 LÉDA (Logistic of Differential Algebraic Equations) and Maths AM-Sud, project 13MATH-06 SIMCA (Implicit Systems, Modeling and Automatic Control).

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Dedicated to the memory of Harold W. Kuhn (1925–2014).

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Ollivier, F. Jacobi’s Bound: Jacobi’s results translated in Kőnig’s, Egerváry’s and Ritt’s mathematical languages. AAECC 34, 793–885 (2023). https://doi.org/10.1007/s00200-022-00547-6

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