(I): Factorization of Linear Ordinary Differential Equations

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1 (I): Factorization of Linear Ordinary Differential Equations Sergey P. Tsarev TU-Berlin, Germany & Krasnoyarsk SPU, Russia

2 Outline A teaser: Landau theorem, 1902 Topics to be discussed today Loewy-Ore theory (and other flavours of factorization) Classical algorithm of factorization (Beke)

3 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators)

4 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators) Example: D 2 = D D

5 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators) Example: D 2 = D D = ( ) ( ) D + 1 x c D 1 x c, c = const.

6 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators) Example: D 2 = D D = ( ) ( ) D + 1 x c D 1 x c, c = const. Landau E. (1902): all possible factorizations of a given operator L into irreducible factors have the same number of factors L = L 1 L k = L 1 L r

7 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators) Example: D 2 = D D = ( ) ( ) D + 1 x c D 1 x c, c = const. Landau E. (1902): all possible factorizations of a given operator L into irreducible factors have the same number of factors L = L 1 L k = L 1 L r = k = r

8 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators) Example: D 2 = D D = ( ) ( ) D + 1 x c D 1 x c, c = const. Landau E. (1902): all possible factorizations of a given operator L into irreducible factors have the same number of factors L = L 1 L k = L 1 L r = k = r and the factors L s, L p are pairwise similar,

9 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators) Example: D 2 = D D = ( ) ( ) D + 1 x c D 1 x c, c = const. Landau E. (1902): all possible factorizations of a given operator L into irreducible factors have the same number of factors L = L 1 L k = L 1 L r = k = r and the factors L s, L p are pairwise similar, in particular the corresponding operators have the same order.

10 A teaser: Landau theorem, 1902 Linear ordinary differential (difference) operators: L = a 0 (x)d n + a 1 (x)d n a n (x), D = d/dx (or D =shift operator), a i Q(x) (or a i k, char k = 0). Usually a 0 1 (monic operators) Example: D 2 = D D = ( ) ( ) D + 1 x c D 1 x c, c = const. Landau E. (1902): all possible factorizations of a given operator L into irreducible factors have the same number of factors L = L 1 L k = L 1 L r = k = r and the factors L s, L p are pairwise similar, in particular the corresponding operators have the same order. Beke E. (1894) gave an algorithm for factorization for the case a i Q(x).

11 1890s 1930s: G. Frobenius, L. Fuchs, L. Heffter, A. Loewy, Ø. Ore...

12 1890s 1930s: G. Frobenius, L. Fuchs, L. Heffter, A. Loewy, Ø. Ore s: M. Bronstein, M. van Hoeij, M. Petkovšek (difference case), F. Schwarz, S. Ts....

13 Outline A teaser: Landau theorem, 1902 Topics to be discussed today Loewy-Ore theory (and other flavours of factorization) Classical algorithm of factorization (Beke)

14 Topics to be (or not to be... ) discussed today:

15 Topics to be (or not to be... ) discussed today: (I) Elementary algebraic theory (Loewy-Ore theory)

16 Topics to be (or not to be... ) discussed today: (I) Elementary algebraic theory (Loewy-Ore theory) (II) Different theoretical flavours / formulations / algorithms for factorization of LODEs

17 Topics to be (or not to be... ) discussed today: (I) Elementary algebraic theory (Loewy-Ore theory) (II) Different theoretical flavours / formulations / algorithms for factorization of LODEs (III) Classical algorithm of factorization (Beke)

18 Topics to be (or not to be... ) discussed today: (I) Elementary algebraic theory (Loewy-Ore theory) (II) Different theoretical flavours / formulations / algorithms for factorization of LODEs (III) Classical algorithm of factorization (Beke) NOT discussed: factorization (decomposition) of nonlinear algebraic ODEs P(x, y(x), y (x),..., y (n) (x)) = 0:

19 Topics to be (or not to be... ) discussed today: (I) Elementary algebraic theory (Loewy-Ore theory) (II) Different theoretical flavours / formulations / algorithms for factorization of LODEs (III) Classical algorithm of factorization (Beke) NOT discussed: factorization (decomposition) of nonlinear algebraic ODEs P(x, y(x), y (x),..., y (n) (x)) = 0: P = Q(x, z, z,..., z (k) ), Q = R(x, y, y,..., y (n k) ).

20 Topics to be (or not to be... ) discussed today: (I) Elementary algebraic theory (Loewy-Ore theory) (II) Different theoretical flavours / formulations / algorithms for factorization of LODEs (III) Classical algorithm of factorization (Beke) NOT discussed: factorization (decomposition) of nonlinear algebraic ODEs P(x, y(x), y (x),..., y (n) (x)) = 0: P = Q(x, z, z,..., z (k) ), Q = R(x, y, y,..., y (n k) ). Algorithms for decomposition: M.Sosnin (2001), Gao et. al. (2003).

21 Outline A teaser: Landau theorem, 1902 Topics to be discussed today Loewy-Ore theory (and other flavours of factorization) Classical algorithm of factorization (Beke)

22 Loewy-Ore theory Loewy, A. Über reduzible lineare homogene Differentialgleichungen. Math. Annalen ( ). Ore, O. Theory of non-commutative polynomials. Annals of Mathematics 34 (1933). N. Jacobson The theory of rings 1943.

23 Loewy-Ore theory Loewy, A. Über reduzible lineare homogene Differentialgleichungen. Math. Annalen ( ). Ore, O. Theory of non-commutative polynomials. Annals of Mathematics 34 (1933). N. Jacobson The theory of rings Right (left) division: for any LODOs L, M, there exist unique LODOs Q, R, Q 1, R 1, such that: L = Q M + R, L = M Q 1 + R 1.

24 Loewy-Ore theory Loewy, A. Über reduzible lineare homogene Differentialgleichungen. Math. Annalen ( ). Ore, O. Theory of non-commutative polynomials. Annals of Mathematics 34 (1933). N. Jacobson The theory of rings Right (left) division: for any LODOs L, M, there exist unique LODOs Q, R, Q 1, R 1, such that: L = Q M + R, L = M Q 1 + R 1. = right (left) GCDs and LCMs: rgcd(l, M) = G L = L 1 G, M = M 1 G (the order of G is maximal);

25 Loewy-Ore theory Loewy, A. Über reduzible lineare homogene Differentialgleichungen. Math. Annalen ( ). Ore, O. Theory of non-commutative polynomials. Annals of Mathematics 34 (1933). N. Jacobson The theory of rings Right (left) division: for any LODOs L, M, there exist unique LODOs Q, R, Q 1, R 1, such that: L = Q M + R, L = M Q 1 + R 1. = right (left) GCDs and LCMs: rgcd(l, M) = G L = L 1 G, M = M 1 G (the order of G is maximal); rlcm(l, M) = K K = M L = L M (the order of K is minimal).

26 Loewy-Ore theory Loewy, A. Über reduzible lineare homogene Differentialgleichungen. Math. Annalen ( ). Ore, O. Theory of non-commutative polynomials. Annals of Mathematics 34 (1933). N. Jacobson The theory of rings Right (left) division: for any LODOs L, M, there exist unique LODOs Q, R, Q 1, R 1, such that: L = Q M + R, L = M Q 1 + R 1. = right (left) GCDs and LCMs: rgcd(l, M) = G L = L 1 G, M = M 1 G (the order of G is maximal); rlcm(l, M) = K K = M L = L M (the order of K is minimal). Exercise: describe algorithms for GCDs and LCMs.

27 Relation to solution spaces:

28 Relation to solution spaces: rgcd(l, M) = G is nontrivial (G 1) Sol(L) Sol(M) {0}

29 Relation to solution spaces: rgcd(l, M) = G is nontrivial (G 1) Sol(L) Sol(M) {0} In general, Sol(L) Sol(M) = Sol(G).

30 Relation to solution spaces: rgcd(l, M) = G is nontrivial (G 1) Sol(L) Sol(M) {0} In general, Sol(L) Sol(M) = Sol(G). rlcm(l, M) = K Sol(L), Sol(M) = {u + v Lu = 0, Mv = 0} = Sol(K )

31 Operator equations: X L + Y M = B, L Z + M T = C with unknown operators X, Y, Z, T are solvable iff respectively rgcd(l, M) divides B on the right and lgcd(l, M) divides C on the left.

32 Def. LODO L is (right) transformed into L 1 by an operator (B is not necessarily monic), and we ll write L B L 1

33 Def. LODO L is (right) transformed into L 1 by an operator (B is not necessarily monic), and we ll write L B L 1 iff rgcd(l, B) = 1 and rlcm(l, B) = L 1 B = B 1 L for some B 1.

34 Def. LODO L is (right) transformed into L 1 by an operator (B is not necessarily monic), and we ll write L B L 1 iff rgcd(l, B) = 1 and rlcm(l, B) = L 1 B = B 1 L for some B 1. = any solution of Lu = 0 is mapped by B into a solution v = Bu of L 1 v = 0.

35 Def. LODO L is (right) transformed into L 1 by an operator (B is not necessarily monic), and we ll write L B L 1 iff rgcd(l, B) = 1 and rlcm(l, B) = L 1 B = B 1 L for some B 1. = any solution of Lu = 0 is mapped by B into a solution v = Bu of L 1 v = 0. One may find with rational algebraic & diff. operations an C operator C such that L 1 L, C B = 1(modL), B C = 1(modL 1 ).

36 Def. LODO L is (right) transformed into L 1 by an operator (B is not necessarily monic), and we ll write L B L 1 iff rgcd(l, B) = 1 and rlcm(l, B) = L 1 B = B 1 L for some B 1. = any solution of Lu = 0 is mapped by B into a solution v = Bu of L 1 v = 0. One may find with rational algebraic & diff. operations an C operator C such that L 1 L, C B = 1(modL), B C = 1(modL 1 ). Operators L, L 1 will be also called similar or of the same kind (in the given differential field k ). They have equal orders.

37 Def. LODO L is (right) transformed into L 1 by an operator (B is not necessarily monic), and we ll write L B L 1 iff rgcd(l, B) = 1 and rlcm(l, B) = L 1 B = B 1 L for some B 1. = any solution of Lu = 0 is mapped by B into a solution v = Bu of L 1 v = 0. One may find with rational algebraic & diff. operations an C operator C such that L 1 L, C B = 1(modL), B C = 1(modL 1 ). Operators L, L 1 will be also called similar or of the same kind (in the given differential field k ). They have equal orders. = For similar operators the problem of solution of the corresponding LODE s Lu = 0, L 1 v = 0 are equivalent.

38 Def. LODO L is (right) transformed into L 1 by an operator (B is not necessarily monic), and we ll write L B L 1 iff rgcd(l, B) = 1 and rlcm(l, B) = L 1 B = B 1 L for some B 1. = any solution of Lu = 0 is mapped by B into a solution v = Bu of L 1 v = 0. One may find with rational algebraic & diff. operations an C operator C such that L 1 L, C B = 1(modL), B C = 1(modL 1 ). Operators L, L 1 will be also called similar or of the same kind (in the given differential field k ). They have equal orders. = For similar operators the problem of solution of the corresponding LODE s Lu = 0, L 1 v = 0 are equivalent. Q.: How one can find out if two given LODOs are similar?

39 Ring-theoretic interpretation: Euclid algorithm = the ring k[d] of LODOs is left-principal and right-principal (no nontrivial two-sided ideals!).

40 Ring-theoretic interpretation: Euclid algorithm = the ring k[d] of LODOs is left-principal and right-principal (no nontrivial two-sided ideals!). = Landau theorem.

41 Ring-theoretic interpretation: Euclid algorithm = the ring k[d] of LODOs is left-principal and right-principal (no nontrivial two-sided ideals!). = Landau theorem. L k[d] generates the left ideal L ;

42 Ring-theoretic interpretation: Euclid algorithm = the ring k[d] of LODOs is left-principal and right-principal (no nontrivial two-sided ideals!). = Landau theorem. L k[d] generates the left ideal L ; L 1 divides L on the right L L 1

43 Ring-theoretic interpretation: Euclid algorithm = the ring k[d] of LODOs is left-principal and right-principal (no nontrivial two-sided ideals!). = Landau theorem. L k[d] generates the left ideal L ; L 1 divides L on the right L L 1 Landau theorem: for L = L 1 L k = L 1 L r we have two maximal chains of ascending left principal ideals L L 2 L k L 3 L k... L k 1 = k[d] and L L 2 L r L 3 L r... L r 1 = k[d] = k = r

44 Ring-theoretic interpretation: Euclid algorithm = the ring k[d] of LODOs is left-principal and right-principal (no nontrivial two-sided ideals!). = Landau theorem. L k[d] generates the left ideal L ; L 1 divides L on the right L L 1 Landau theorem: for L = L 1 L k = L 1 L r we have two maximal chains of ascending left principal ideals L L 2 L k L 3 L k... L k 1 = k[d] and L L 2 L r L 3 L r... L r 1 = k[d] = k = r Jordan-Hölder theorem

45 Lattice-theoretic interpretation:

46 Lattice-theoretic interpretation: Partially ordered by inclusion set M (called a poset) of (left) ideals in k[d] has the following two fundamental properties:

47 Lattice-theoretic interpretation: Partially ordered by inclusion set M (called a poset) of (left) ideals in k[d] has the following two fundamental properties: a) for any two elements A, B M (left ideals!) one can find a unique C = sup(a, B), i.e. such C that C A, C B, and C is minimal possible.

48 Lattice-theoretic interpretation: Partially ordered by inclusion set M (called a poset) of (left) ideals in k[d] has the following two fundamental properties: a) for any two elements A, B M (left ideals!) one can find a unique C = sup(a, B), i.e. such C that C A, C B, and C is minimal possible. Analogously there exist a unique D = inf(a, B), D A, D B, D is maximal possible.

49 Lattice-theoretic interpretation: Partially ordered by inclusion set M (called a poset) of (left) ideals in k[d] has the following two fundamental properties: a) for any two elements A, B M (left ideals!) one can find a unique C = sup(a, B), i.e. such C that C A, C B, and C is minimal possible. Analogously there exist a unique D = inf(a, B), D A, D B, D is maximal possible. Such posets are called lattices. sup(a, B) and inf(a, B) correspond to the GCD and the LCM.

50 Lattice-theoretic interpretation: Partially ordered by inclusion set M (called a poset) of (left) ideals in k[d] has the following two fundamental properties: a) for any two elements A, B M (left ideals!) one can find a unique C = sup(a, B), i.e. such C that C A, C B, and C is minimal possible. Analogously there exist a unique D = inf(a, B), D A, D B, D is maximal possible. Such posets are called lattices. sup(a, B) and inf(a, B) correspond to the GCD and the LCM. For simplicity (and following the established tradition) sup(a, B) will be hereafter denoted as A + B and inf(a, B) as A B;

51 Lattice-theoretic interpretation: Partially ordered by inclusion set M (called a poset) of (left) ideals in k[d] has the following two fundamental properties: a) for any two elements A, B M (left ideals!) one can find a unique C = sup(a, B), i.e. such C that C A, C B, and C is minimal possible. Analogously there exist a unique D = inf(a, B), D A, D B, D is maximal possible. Such posets are called lattices. sup(a, B) and inf(a, B) correspond to the GCD and the LCM. For simplicity (and following the established tradition) sup(a, B) will be hereafter denoted as A + B and inf(a, B) as A B; b) For any three A, B, C M the following modular identity holds: (A C + B) C = A C + B C

52 Lattice-theoretic interpretation: Partially ordered by inclusion set M (called a poset) of (left) ideals in k[d] has the following two fundamental properties: a) for any two elements A, B M (left ideals!) one can find a unique C = sup(a, B), i.e. such C that C A, C B, and C is minimal possible. Analogously there exist a unique D = inf(a, B), D A, D B, D is maximal possible. Such posets are called lattices. sup(a, B) and inf(a, B) correspond to the GCD and the LCM. For simplicity (and following the established tradition) sup(a, B) will be hereafter denoted as A + B and inf(a, B) as A B; b) For any three A, B, C M the following modular identity holds: (A C + B) C = A C + B C Such lattices are called modular lattices or Dedekind structures.

53 Modularity =

54 Modularity = (Jordan-Hölder-Dedekind chain condition): any two finite maximal chains L > L 1 > > L k > 0 L > M 1 > > M r > 0 for a given L M have equal lengths: k = r (the same for ascending chains).

55 Modularity = (Jordan-Hölder-Dedekind chain condition): any two finite maximal chains L > L 1 > > L k > 0 L > M 1 > > M r > 0 for a given L M have equal lengths: k = r (the same for ascending chains). There are notions of similarity, direct sums, Kurosh & Ore theorems on direct sums...

56 Categorical interpretation: abelian category of LODEs

57 Categorical interpretation: abelian category of LODEs Objects are monic operators L = D n + a 1 (x)d n a n (x), or, equivalently, their solution spaces (finite-dimensional!) (... but these spaces are not constructive... )

58 Categorical interpretation: abelian category of LODEs Objects are monic operators L = D n + a 1 (x)d n a n (x), or, equivalently, their solution spaces (finite-dimensional!) (... but these spaces are not constructive... ) Morphisms are mappings of solutions with auxiliary (non-monic) operators: P : L M iff for every u such that Lu = 0, v = Pu gives a solution of M: Mv = 0. All operators here have coefficients in some fixed diff. field.

59 Categorical interpretation: abelian category of LODEs Objects are monic operators L = D n + a 1 (x)d n a n (x), or, equivalently, their solution spaces (finite-dimensional!) (... but these spaces are not constructive... ) Morphisms are mappings of solutions with auxiliary (non-monic) operators: P : L M iff for every u such that Lu = 0, v = Pu gives a solution of M: Mv = 0. All operators here have coefficients in some fixed diff. field. Two operators P 1, P 2 generate the same morphism P 1 = P 2 (mod L).

60 (!) This definition is NOT equivalent to the definition of transformation of operators: Why? L P M

61 (!) This definition is NOT equivalent to the definition of transformation of operators: Why? Because for morphisms: L P M 1) P and L may have common solutions, i.e. nontrivial rgcd(p, L). This means that the mapping of the solution space Sol(L) by P may have a kernel Sol(rGCD(P, L)). The morphism is not injective in this case.

62 (!) This definition is NOT equivalent to the definition of transformation of operators: Why? Because for morphisms: L P M 1) P and L may have common solutions, i.e. nontrivial rgcd(p, L). This means that the mapping of the solution space Sol(L) by P may have a kernel Sol(rGCD(P, L)). The morphism is not injective in this case. 2) The image of the solution space Sol(L) may be smaller than Sol(M). The morphism is not surjective in this case.

63 (!) This definition is NOT equivalent to the definition of transformation of operators: Why? Because for morphisms: L P M 1) P and L may have common solutions, i.e. nontrivial rgcd(p, L). This means that the mapping of the solution space Sol(L) by P may have a kernel Sol(rGCD(P, L)). The morphism is not injective in this case. 2) The image of the solution space Sol(L) may be smaller than Sol(M). The morphism is not surjective in this case. Algebraically: M P = N L for some N,

64 (!) This definition is NOT equivalent to the definition of transformation of operators: Why? Because for morphisms: L P M 1) P and L may have common solutions, i.e. nontrivial rgcd(p, L). This means that the mapping of the solution space Sol(L) by P may have a kernel Sol(rGCD(P, L)). The morphism is not injective in this case. 2) The image of the solution space Sol(L) may be smaller than Sol(M). The morphism is not surjective in this case. Algebraically: M P = N L for some N, M P = N L rlcm(l, P).

65 Fact: This category of LODO is abelian.

66 Fact: This category of LODO is abelian. Theorem Any abelian category with finite ascending chains satisfies the Jordan-Hölder property (= Landau theorem).

67 Fact: This category of LODO is abelian. Theorem Any abelian category with finite ascending chains satisfies the Jordan-Hölder property (= Landau theorem). Again: there are notions of similarity, direct sums, Kurosh & Ore theorems on direct sums, homological algebra...

68 Fact: This category of LODO is abelian. Theorem Any abelian category with finite ascending chains satisfies the Jordan-Hölder property (= Landau theorem). Again: there are notions of similarity, direct sums, Kurosh & Ore theorems on direct sums, homological algebra... Q.: Why do we need this abstract nonsense??

69 Fact: This category of LODO is abelian. Theorem Any abelian category with finite ascending chains satisfies the Jordan-Hölder property (= Landau theorem). Again: there are notions of similarity, direct sums, Kurosh & Ore theorems on direct sums, homological algebra... Q.: Why do we need this abstract nonsense?? A.: this is really needed for the case of LPDEs!

70 Outline A teaser: Landau theorem, 1902 Topics to be discussed today Loewy-Ore theory (and other flavours of factorization) Classical algorithm of factorization (Beke)

71 Classical algorithm of factorization (Beke) and its modern rivals Ideas:

72 Classical algorithm of factorization (Beke) and its modern rivals Ideas: (I) L = L 1 (D u(x)) u = y y, y(x) is a solution of Ly = 0. For u k = Q(x) this means y = exp( u dx) a hyperexponential solution of Ly = 0.

73 Classical algorithm of factorization (Beke) and its modern rivals Ideas: (I) L = L 1 (D u(x)) u = y y, y(x) is a solution of Ly = 0. For u k = Q(x) this means y = exp( u dx) a hyperexponential solution of Ly = 0. (II) L = L 1 L 2, ord(l 2 ) = m = for some associated LODO L (m), L (m) = L 1 (D u(x))

Algebra Homework, Edition 2 9 September 2010

Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.