1. Jacobi’s work on normal forms of differential systems François Ollivier LIX UMR CNRS-Ecole polytechnique 7161 AMS special session on Differential Algebra, april 15 th 2007

2. Jacobi’s bound A set of manuscripts, written around 1836. Two posthumous papers 1865, 1866. u 1 (x)=0, …, u n (x) = 0 differential system in x 1, …, x n A bound: J  max σ  Sn  i ord x σ(i) u i (maximal transversal sum). The truncated determinant condition  0  ord(u)  J. An algorithm to compute J in polynomial time A method to compute normal forms using as few derivatives as possible (if  0).

3. Jacobi’s algorithm How to compute the bound without trying n! sums? A canon is a square matrix that possess maximal element (in their column), located in different rows. Jacobi’s algorithm computes minimal integers λ i to be added to each line of the order matrix in order to make in a canon. It is similar to Kuhn’s Hungarian method (1955). Harold W. Kuhn Professor Emeritus Department of Mathematics Princeton University THE HUNGARIAN METHOD FOR THE ASSIGNMENT PROBLEM AND HOW JACOBI BEAT ME BY 100 YEARS (A conference given in September 2007)

4. Normal form computation Let a i,j :  ord x j u i. Let Λ:  max i λ i, α i :  Λ  λ i and β j :  max i a i,j  α i.  :  (∂u i /∂x j (α i  β i ) ) is the truncated jacobian matrix. It is equal to (∂u i /∂x j (ord x j u i ) ), where the terms such that deg x j u i does not appear in a maximal sum equal to J have been replaced by 0. Theorem. — If |  |  0, then the order is equal to J and it is possible to compute a normal form (a characteristic set), using only derivatives of equation u i up to λ i For generic systems, we need go up to λ i. If |  |  0, then the order is < J and we need orders greater than λ i for generic systems.

5. Proof Assume that λ 1 ≥ … ≥ λ n and that the n principal minors of  are all of full rank. Take Jacobi’s ordering defined by x j (k) < x j ’(k’) if k-β j < k-β j’ or if k-β j = k-β j’ and j>j’. Theorem. — [u]:|  | ∞ is radical and we can compute a char. set. for this ordering using derivatives of u i up to λ i. We use “Lazard’s lemma”, remarking that for all 1≤k≤n the jacobian matrix of u 1 (λ 1 -λ k ), …, u k with respect to x 1 (α k  β 1 ), …, x n (α k  β n ) corresponds to the first k rows of . u 1 is candidate to be the first element A 1 of the char. set A. Reduce u k by the first k  1 candidate  candidate for A k. Proceed as in Boulier 1995, Hubert, etc… We get a “regular representation”.

6. Resolvent computation Assume that |  |  0 and that x j is a primitive element for all component of [u]:|  | ∞. Theorem. — We can compute a resolvent representation P(x j )=0, x k =R k (x j ), k  j, of [u]:|  | ∞ using only derivatives of u i up to order μ i,j. Take (a i,j ), suppress row i and column j, μ i,j is the maximal transversal sum of this matrix. Proof rely on combinatorial results, close to Jacobi’s algorithm. Start with a char. set A for Jacobi’s ordering, count how much we need to differentiate each element to compute derivatives of x j up to order J.

7. Normal forms of a system of 2 variables Jacobi’s problem: x 1 (e 1 ) = P 1 (x 1, x 2 ), x 2 (e 2 ) = P 2 (x 1, x 2 ), how to lower the order e 1 ? If ord x 1 P 2 = f 1, there exists a normal form x 1 (f 1 ) = Q 1 (x 1, x 2 ), x 2 (f 2 ) = P 2 (x 1, x 2 ), and no normal form with order f 1 <g 1 < e 1. Same for char. sets. For any set I  [1,J], there exists a system such that I is the set of orders of x 1 in all its possible normal forms (char. sets).

8. Normal forms general case Jacobi’s problem: x j (e j ) = P j (x), 1j, how to lower the orders e 1, …, e r ? If 2r≤n, ord x i P r+1, …, P n = f j, and the rank of the jacobian matrix of P r+1, …, P n with respect to x 1, …, x r is full, it works. Same for char. sets. If not, apply Jacobi’s method for system P r+1, …, P n. Former introduction of first paper (S. Cohn’s transcription), suppressed by Borchardt. What are the possible n-uple of orders for sets of normal forms?

9. Conclusion Tam quaestiones altioris indaginis poscuntur. Under a genericity hypothesis often encountered in practice, we can solve a system very fast. It may be possible to avoid the elimination and use implicit systems (Jacobi’s normal forms were implicit). Possible fast computation of resolvents, important for methods relying on the TERA philosophy (represent polynomials by s. l. programs computing them). Giusti, Heintz, … in the 1990 ies for algebraic syst., D’Alfonso 2006, diff. case.

10. http://www.lix.polytechnique.fr/~ollivier/JACOBI/jacobiEngl.htm Translations of the papers available in French and (for some of them) in English. Transcriptions and translations in progress… look for updates.