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Jacobi’s work on normal forms of differential systems

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. Jacobi’s bound. A set of manuscripts, written around 1836. Two posthumous papers 1865, 1866.

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Jacobi’s work on normal forms of differential systems

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  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 15th 2007

  2. Jacobi’s bound • A set of manuscripts, written around 1836. • Two posthumous papers 1865, 1866. u1(x)=0, …, un(x) = 0 differential system in x1, …, xn • A bound: J:=maxσSni ord xσ(i)ui (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. How to compute the bound without trying n! sums? Acanon is a square matrix that possess maximal element (in their column), located in different rows. Jacobi’s algorithm computes minimal integers λito 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 EmeritusDepartment of MathematicsPrinceton University THE HUNGARIAN METHOD FOR THE ASSIGNMENT PROBLEMAND HOW JACOBI BEAT ME BY 100 YEARS (A conference given in September 2007) Jacobi’s algorithm

  4. Let ai,j :ordxjui. Let Λ:maxi λi, αi:Λ λi and βj:maxi ai,j  αi. : (∂ui /∂xj(αiβi)) is thetruncated jacobian matrix. It is equal to (∂ui /∂xj(ordxjui)), where the terms such that degxjui does not appear in a maximal sum equal to J have been replaced by 0.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 ui up to λi For generic systems, we need go up to λi. If ||0, then the order is <J and we need orders greater than λi for generic systems. Normal form computation

  5. Assume that λ1≥ …≥ λn and that the n principal minors of  are all of full rank. Take Jacobi’s ordering defined byxj(k) < xj’(k’) if k-βj < k-βj’ or if k-βj = k-βj’ and j>j’. Theorem.— [u]:||∞ is radical and we can compute a char. set. for this ordering using derivatives of ui up to λi. We use “Lazard’s lemma”, remarking that for all 1≤k≤n the jacobian matrix of u1(λ1-λk),…,uk with respect to x1(αkβ1), …, xn(αkβn) corresponds to the first k rows of . u1is candidate to be the first element A1 of the char. set A. Reduce uk by the first k1 candidate  candidate for Ak. Proceed as in Boulier 1995, Hubert, etc… We get a “regular representation”. Proof

  6. Resolvent computation Assume that ||0 and that xjis a primitive element for all component of [u]:||∞. Theorem. — We can compute a resolvent representation P(xj)=0, xk=Rk(xj), kj, of [u]:||∞ using only derivatives of ui up to order μi,j . Take (ai,j), suppress row i and column j, μi,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 xj up to order J.

  7. Normal forms of a system of 2 variables Jacobi’s problem:x1 (e1)= P1(x1, x2), x2 (e2)= P2(x1, x2), how to lower the order e1? If ordx1P2 = f1, there exists a normal form x1 (f1)= Q1(x1, x2), x2 (f2)= P2(x1, x2), and no normal form with order f1 <g1< e1. Same for char. sets. For any set I[1,J], there exists a system such that I is the set of orders of x1 in all its possible normal forms (char. sets).

  8. Normal forms general case Jacobi’s problem:xj (ej)= Pj(x), 1j, how to lower the orders e1 , …,er? If 2r≤n, ordxiPr+1 , …, Pn = fj, and the rank of the jacobian matrix of Pr+1 , …, Pnwith respect to x1 , …,xr is full, it works. Same for char. sets. If not, apply Jacobi’s method for system Pr+1 , …, Pn. 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 1990ies for algebraic syst., D’Alfonso 2006, diff. case.

  10. http://www.lix.polytechnique.fr/~ollivier/JACOBI/jacobiEngl.htmhttp://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.

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