Sparse Differential Resultant for Laurent Differential Polynomials. Wei Li

Transcription

1 Sparse Differential Resultant for Laurent Differential Polynomials Wei Li Academy of Mathematics and Systems Science Chinese Academy of Sciences KSDA 2011 Joint work with X.S. Gao and C.M. Yuan Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

2 Motivations Sparse Resultant is a basic concept in algebraic geometry and a powerful tool in algebraic elimination theory with important applications. Differential polynomials from practice are usually sparse, and their differential resultant may vanish identically. Sparse differential resultant is not studied before. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

3 Work on Sparse Resultant Gelfand, Kapranov, and Zelevinsky (1991, 1994) introduced the sparse resultant. Sturmfels (1993, 1994) proved basic properties for the sparse resultant. Canny and Emiris (1993, 1995, 2000) gave matrix formulas for sparse resultants and efficient algorithms. D Andrea (2002) proved a sparse resultant is the quotient of two determinants. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

4 Work on Differential Resultant Ritt (1932): Differential resultant for two differential polynomials in one diff indeterminate. Ferro (1997): Differential resultant as algebraic Macaulay resultant. Ferrro s resultant of two generic diff polynomials of degree larger than one is always zero. Chardin (1991): Resultant for differential operators. Rueda-Sendra (2010): Differential resultant of a linear system. Gao, Li, Yuan (2010): Rigorous definition of differential resultant of n + 1 differential polynomials in n indeterminates. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

5 Outline of the Talk Sparse diff resultant for Laurent diff polynomials Properties of sparse diff resultant Criterion for Laurent diff essential system in terms of supports A single exponential algorithm to compute sparse diff resultant Sparse diff resultant for diff polynomial with non-vanishing degree zero terms Summary Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

6 Sparse Differential Resultant for Laurent Differential Polynomials Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

7 Notations Ordinary differential field: (F, δ). e.g. (Q(x), d dx ) Universal differential field of F: (E, δ). Diff Indeterminates: Y = {y 1,..., y n }. Notation: y (k) i = δ k y i, Y [t] = {y (k) : k t}. Differential Monomial: M = n k=1 m: set of diff monomials in Y. i o (l) l=0 (y k )d kl with d kl Z 0 ; Differential polynomial ring: F{Y} = F[y (k) i : k 0]. Notations: For S F{Y}, [S] = the diff ideal generated by S [S] : m = {f F{Y} M m, s.t. M f [S]}. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

8 Laurent Differential Polynomial Ring Laurent Diff Monomial: M = n o (l) k=1 l=0 (y k )d kl with d kl Z; Laurent Diff Polynomial: f = m k=1 a km k, M k Laurent diff monomials. Support of f : A = {M 1,..., M m }. Norm Form of f : f N = Mf, where M is the denominator of f. Laurent Diff Polynomials Ring: F{Y, Y 1 }. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

9 Laurent Differential Polynomial Ring Laurent Diff Monomial: M = n o (l) k=1 l=0 (y k )d kl with d kl Z; Laurent Diff Polynomial: f = m k=1 a km k, M k Laurent diff monomials. Support of f : A = {M 1,..., M m }. Norm Form of f : f N = Mf, where M is the denominator of f. Laurent Diff Polynomials Ring: F{Y, Y 1 }. E : universal diff field of F. E = E\{a E : k 0 s.t. a (k) = 0}. Non-polynomial Diff Solution: ξ (E ) n s.t. f (ξ) = 0. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

10 Sparse Laurent Diff Polynomials A = {M 1,..., M m }: finite set of Laurent Diff monomials. Notation: L(A) = { m k=1 b km k : b k E}. Sparse Laurent Diff Polynomial w.r.t. A: f L(A). Generic Sparse Laurent Diff Polynomial w.r.t. A: P L(A) with coefficients diff independent over Q. Given A i (i = 0,..., n) and f i L(A i ), we ask: f i have a common non-polynomial solution? Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

11 Sparse Differential Resultant Generic Sparse Laurent Diff polynomials w.r.t. A i : P i = l i j=0 u ijm ij L(A i ) with coefficients u i. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

12 Sparse Differential Resultant Generic Sparse Laurent Diff polynomials w.r.t. A i : P i = l i j=0 u ijm ij L(A i ) with coefficients u i. Sparse Differential Resultant Exists: [P 0,..., P n ] Q{u 0, u 1,..., u n } = sat(r(u 0,..., u n )) is of codimension 1 Definition R is defined to be the Sparse Differential Resultant of P i, denoted by Res P0,...,P n, or Res A0,...,A n. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

13 Sparse Differential Resultant Generic Sparse Laurent Diff polynomials w.r.t. A i : P i = l i j=0 u ijm ij L(A i ) with coefficients u i. Sparse Differential Resultant Exists: [P 0,..., P n ] Q{u 0, u 1,..., u n } = sat(r(u 0,..., u n )) is of codimension 1 P i are Laurent differentially essential: There exist k i (i = 0,..., n) with 1 k i l i such that d.tr.deg Q M 0k 0 M 00, M 1k 1 M 10,..., M nkn M n0 /Q = n. Definition R is defined to be the Sparse Differential Resultant of P i, denoted by Res P0,...,P n, or Res A0,...,A n. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

14 Facts: Laurent diff essential does not depend on M i0 : Lemma There exist (k i, j i ) with k i j i s.t. d.tr.deg Q M 0k 0 M 0j0,..., M nkn /Q = n. There exist k i with 1 k i l i s.t. d.tr.deg Q M 0k 0 M 00,..., M nkn M n0 /Q = n. M nj n Definition for Sparse Diff Resultant relies on the fact: Theorem [P N 0,..., PN n ] : m is a prime differential ideal in Q{Y, u 0,..., u n }. ([P N 0,..., PN n ] : m) Q{u 0,..., u n } is of codimension 1 if and only if P 0,..., P n are Laurent diff essential. [P 0,..., P n ] Q{u 0,..., u n } = ([P N 0,..., PN n ] : m) Q{u 0,..., u n }. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

15 A Property on Differential Specialization Key fact used in the proof: Theorem U = (u 1,..., u r ) and Y = (y 1,..., y n ): sets of diff indeterminates. P i (U, Y) F Y {U} (i = 1,..., m). If P i (U, Y) are diff dependent over F U, then for any specialization U to U F over F, P i (U, Y) are diff dependent over F. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

16 Examples Example (1) P 0 = u 00 + u 01 y 1 y 2 P 1 = u 10 + u 01 y 1 y 2 P 2 = u 20 + u 21 y 1 y 2. P 0, P 1, P 2 form a Laurent diff essential system. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

17 Examples Example (1) P 0 = u 00 + u 01 y 1 y 2 P 1 = u 10 + u 01 y 1 y 2 P 2 = u 20 + u 21 y 1 y 2. P 0, P 1, P 2 form a Laurent diff essential system. Using diff characteristic set method, we can compute Res P0,P 1,P 2 = u 11 u 2 20 u2 01 u 01u 00 u 2 21 u 10 + u 01 u 11 u 20 u 21 u 00 u 11u 20 u 00 u 21 u 01. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

18 Examples Example (2) n = 1 and A 0 = A 1 = {y, y, y 2 }. A 0, A 1 are Laurent diff essential. Res A = u 12 u 01 u 00 u 10 u 12 u 2 01 u 10 + u 12u 01 u 11 u 00 + u 12 u 01 u 11 u 00 u 11 u 02 u 00 u 10 + u 11 u 02 u 10 u 01 + u 02 u 01 u 2 10 u2 11 u 02u 00 +u 11 u 02 u 01 u 10 + u 11 u 2 00 u 12 + u 2 11 u 02 u 00 u 11 u 02 u 01u 10 u 11 u 01 u 12 u 00 + u 2 01 u 12 u 10 u 11 u 01 u 12u 00 u 11 u 02u 01 u 10. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

19 Basic Properties of Sparse Differential Resultant Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

20 Properties of Sparse Differential Resultant Theorem (Order and Diff homogeneity) Diff resultant is diff homogeneous in each u i and is of order h i = s s i in u i (i = 0,..., n) where s = n l=0 s l. Sparse diff resultant is diff homogeneous in each u i and is of order h i s s i in u i. Degree Bound will be given later! Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

21 Conditions for Existence of Non-polynomial Solutions A 0,..., A n : Laurent diff essential; (F 0,..., F n ) L(A 0 ) L(A n ): F i = l i j=0 v ijm ij. Z 0 (A 0,..., A n ): set of F i having a common non-polynomial solution. Z(A 0,..., A n ): Kolchin diff closure of Z 0 (A 0,..., A n ). Theorem Z(A 0,..., A n ) = V ( sat(res A0,...,A n ) ). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

22 Conditions for Existence of Non-polynomial Solutions A 0,..., A n : Laurent diff essential; (F 0,..., F n ) L(A 0 ) L(A n ): F i = l i j=0 v ijm ij. Z 0 (A 0,..., A n ): set of F i having a common non-polynomial solution. Z(A 0,..., A n ): Kolchin diff closure of Z 0 (A 0,..., A n ). Theorem Z(A 0,..., A n ) = V ( sat(res A0,...,A n ) ). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

23 Example The need to consider non-polynomial solutions: Example n = 2, and (F, δ) = (Q(x), d dx ). P i = u i0 y 1 + u i1y 1 + u i2y 2 (i = 0, 1, 2). P i form a Laurent diff essential system for d.tr.deg Q y 1 y 1 The sparse differential resultant is u 00 u 01 u 02 Res P0,P 1,P 2 = u 10 u 11 u 12 u 20 u 21 u 22 0., y 2 /Q = 2. y 1 But ξ = (c 11 x + c 10, c 22 x 2 + c 21 x + c 20 ) / (E ) 2 is a non-zero solution of P i = 0 (i = 0, 1, 2) where c ij are distinct arbitrary constants. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

24 Condition for Existence of Unique Non-polynomial Solutions Assume 1) Any n of the A i diff independent: ord(r, u i ) 0 for each i. 2) e j Span Z {α ij α i0 }. Theorem (P i, u i ) specialize to (P i, v i ). If R(v 0,..., v n ) = 0 and R u (h i ) ik (v 0,..., v n ) 0, then P i = 0 have at most a unique common non-polynomial solution. Z 1 (A 0,..., A n ): set of (v 0,..., v n ) s.t. P i have a unique non-polynomial solution. Then Z 1 (A 0,..., A n ) = V ( sat(res A0,...,A n ) ) Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

25 Differential Toric Variety A = {M 0,..., M l }: d.tr.deg Q M 1 M 0,..., M l M 0 /Q = n. P(l): l-dimensional diff projective space. Consider φ A : (E ) n P(l) ξ (M 0 (ξ), M 1 (ξ),..., M l (ξ)) Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

26 Differential Toric Variety A = {M 0,..., M l }: d.tr.deg Q M 1 M 0,..., M l M 0 /Q = n. P(l): l-dimensional diff projective space. Consider φ A : (E ) n P(l) ξ (M 0 (ξ), M 1 (ξ),..., M l (ξ)) Definition The Kolchin closure of φ A ( (E ) n) is defined to be the diff toric variety w.r.t. A, denoted by X A. That is, X A = φ A ( (E ) n). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

27 Differential Toric Variety A = {M 0,..., M l }: d.tr.deg Q M 1 M 0,..., M l M 0 /Q = n. P(l): l-dimensional diff projective space. Consider φ A : (E ) n P(l) ξ (M 0 (ξ), M 1 (ξ),..., M l (ξ)) Definition The Kolchin closure of φ A ( (E ) n) is defined to be the diff toric variety w.r.t. A, denoted by X A. That is, X A = φ A ( (E ) n). Theorem X A is an irreducible projective diff variety of dim n over Q. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

28 The Relation of Diff Toric variety and Diff Chow Form Laurent Diff Polynomial: P i = u i0 M 0 + u i1 M u il M l Generic Projective Diff Hyperplane in P(l): L i = u i0 z 0 + u i1 z u il M l Theorem Res A is the differential Chow form of X A. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

29 The Relation of Diff Toric variety and Diff Chow Form Laurent Diff Polynomial: P i = u i0 M 0 + u i1 M u il M l Generic Projective Diff Hyperplane in P(l): L i = u i0 z 0 + u i1 z u il M l Theorem Res A is the differential Chow form of X A. Example (Continue from Example 2) In this example, X A = V(sat(z 1 z 2 (z 0 z 2 z 0 z 2))). And Res A is the diff Chow form of X A. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

30 Poisson-Type Product Formula Algebraic Resultant: Res(A(x), B(x)) = c η,b(η)=0 A(η). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

31 Poisson-Type Product Formula Algebraic Resultant: Res(A(x), B(x)) = c η,b(η)=0 A(η). Differential Resultant: δres(u 0,..., u n ) = A(u 0,..., u n ) t 0 τ=1 P 0 (η τ1,..., η τn ) (h 0). And (η τ1,..., η τn ) are generic points of [P 1,..., P n ]. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

32 Poisson-Type Product Formula Algebraic Resultant: Res(A(x), B(x)) = c η,b(η)=0 A(η). Differential Resultant: δres(u 0,..., u n ) = A(u 0,..., u n ) t 0 τ=1 P 0 (η τ1,..., η τn ) (h 0). And (η τ1,..., η τn ) are generic points of [P 1,..., P n ]. Sparse Differential Resultant: Res(u 0,..., u n ) = A t 0 τ=1 (u 00 + l 0k=1 u 0k ξ τk ) (h 0). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

33 Possion Type Product Formula (Extended) Theorem When 1) Any n of the A i diff independent and 2) e j Span Z {α ij α i0 }, the result can be strengthened: Res(u 0,..., u n ) = A t 0 τ=1 ( P0 (η τ1,...,η τn) M 00(ητ1,...,ητn) ) (h0 ). And η τ = (η τ1,..., η τn ) lie on P 1,..., P n. Furthermore, they are generic points of [P N 0,..., PN n ] : m. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

34 Representation of the Sparse Resultant Algebraic Resultant: Res(A(x), B(x)) = A(x)T(x) + B(x)W(x), where deg(t ) < deg(b), deg(w ) < deg(a). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

35 Representation of the Sparse Resultant Algebraic Resultant: Res(A(x), B(x)) = A(x)T(x) + B(x)W(x), where deg(t ) < deg(b), deg(w ) < deg(a). Differential Resultant: δres(u 0,..., u n ) = n i=0 s si j=0 h ijp (j) i. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

36 Representation of the Sparse Resultant Algebraic Resultant: Res(A(x), B(x)) = A(x)T(x) + B(x)W(x), where deg(t ) < deg(b), deg(w ) < deg(a). Differential Resultant: δres(u 0,..., u n ) = n i=0 s si j=0 h ijp (j) i. Sparse Differential Resultant: Res(u 0,..., u n ) = n hi i=0 j=0 h ij(p N i ) (j), where h i s s i. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

37 Representation of the Sparse Resultant Algebraic Resultant: Res(A(x), B(x)) = A(x)T(x) + B(x)W(x), where deg(t ) < deg(b), deg(w ) < deg(a). Differential Resultant: δres(u 0,..., u n ) = n i=0 s si j=0 h ijp (j) i. Sparse Differential Resultant: Res(u 0,..., u n ) = n hi i=0 j=0 h ij(p N i ) (j), where h i s s i. Degree bound for this linear combination will be given later. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

38 Criterion for Laurent Diff Essential in terms of Supports Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

39 Basic Notions s k=0 B i = n j=1 ) d ijk (i = 1,..., m): Laurent Diff Monomials. New algebraic indeterminates: x 1,..., x n. d ij = s k=0 d ijkxj k (i = 1,..., m; j = 1,..., n). Symbolic Support Vector (SSV) of B i : β i = (d i1,..., d in ). (y (k) j Symbolic Support Matrix (SSM) of B 1,..., B m : β 1 β 2 M =. β m d 11 d d 1n = d 21 d d 0n... d m1 d m2... d mn Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

40 Basic Notions s k=0 B i = n j=1 ) d ijk (i = 1,..., m): Laurent Diff Monomials. New algebraic indeterminates: x 1,..., x n. d ij = s k=0 d ijkxj k (i = 1,..., m; j = 1,..., n). Symbolic Support Vector (SSV) of B i : β i = (d i1,..., d in ). (y (k) j Symbolic Support Matrix (SSM) of B 1,..., B m : β 1 β 2 M =. β m d 11 d d 1n = d 21 d d 0n... d m1 d m2... d mn Def: Call B 1,..., B m Reduced if deg(d ii ) > deg(d ji )(j > i) for each i min(m, n). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

41 Basic Notions s k=0 B i = n j=1 ) d ijk (i = 1,..., m): Laurent Diff Monomials. New algebraic indeterminates: x 1,..., x n. d ij = s k=0 d ijkxj k (i = 1,..., m; j = 1,..., n). Symbolic Support Vector (SSV) of B i : β i = (d i1,..., d in ). (y (k) j Symbolic Support Matrix (SSM) of B 1,..., B m : β 1 β 2 M =. β m d 11 d d 1n = d 21 d d 0n... d m1 d m2... d mn Def: Call B 1,..., B m Reduced if deg(d ii ) > deg(d ji )(j > i) for each i min(m, n). Lemma B 1,..., B m Reduced. Then d.tr.deg Q B 1, B 2,..., B m /Q = min(m, n). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

42 Laurent Diff monomials in T-shape Def: Call B 1,..., B m in T-shape with index (i, j) if there exist (i, j) s.t. 1) (M 1 ) i i and (M 2 ) (n i) j are reduced sub-matrices, 2) Z 1 and Z 2 constitute an (m i) (n j) zero matrix. Figure: T-shape Matrix Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

43 Laurent Diff monomials in T-shape Def: Call B 1,..., B m in T-shape with index (i, j) if there exist (i, j) s.t. 1) (M 1 ) i i and (M 2 ) (n i) j are reduced sub-matrices, 2) Z 1 and Z 2 constitute an (m i) (n j) zero matrix. Figure: T-shape Matrix Theorem B 1,..., B m in T-shape. d.tr.deg Q B 1, B 2,..., B m /Q = rk(m) = i + j. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

44 General Laurent Diff Monmials M: SSM of arbitrary B 1,..., B m. Q-elementary Transformations: row(column) interchanging, and adding a Q-multiple of one row to another. Theorem M can be reduced to a T-shape matrix by finite Q-elementary transformations. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

45 General Laurent Diff Monmials M: SSM of arbitrary B 1,..., B m. Q-elementary Transformations: row(column) interchanging, and adding a Q-multiple of one row to another. Theorem M can be reduced to a T-shape matrix by finite Q-elementary transformations. Q-elementary transformations keep diff transcendence degree: Theorem d.tr.deg Q B 1, B 2,..., B m /Q = rk(m) Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

46 Generic Laurent Diff Polynomials P i = l i j=0 u ijm ij (i = 1,..., m). SSV of M ij /M i0 : β ij Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

47 Generic Laurent Diff Polynomials P i = l i j=0 u ijm ij (i = 1,..., m). SSV of M ij /M i0 : β ij SSV of P i : β i = l i j=0 u ijβ ij. SSM of P 0,..., P n : M P = (β 0,..., β n ) T Theorem 1) d.tr.deg Q û P 1 M 10,..., Pm M m0 /Q û = rk(m P ) (û = m i=1 u i). 2) Codim([P N 1,..., PN m] : m) Q{u 0,..., u n } = n + 1 rk(m P ). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

48 Criterion for Laurent Diff Essential P i = l i j=0 u ijm ij (i = 0,..., n). Theorem The P i are Laurent differentially essential rk(m P ) = n there exist k i (1 k i l i ) s.t. rk(m k0,...,k n ) = n where M k0,...,k n is the SSM for M 0k0 /M 00,..., M nkn /M n,0. Remark: Here, a diff problem is translated to a linear algebraic one. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

49 A Single Exponential Algorithm to Compute Sparse Diff Resultant Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

50 Degree of Sparse Differential Resultant Laurent Diff Essential System: P 0,..., P n ord(p i ) = s i and deg(p i ) = m i. P N i = l j=0 u ijn ij R : the sparse resultant of P 0,..., P n. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

51 Degree of Sparse Differential Resultant Laurent Diff Essential System: P 0,..., P n ord(p i ) = s i and deg(p i ) = m i. P N i = l j=0 u ijn ij R : the sparse resultant of P 0,..., P n. Theorem 1 ord(r, u i ) = h i s s i, where s = n i=0 s i. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

52 Degree of Sparse Differential Resultant Laurent Diff Essential System: P 0,..., P n ord(p i ) = s i and deg(p i ) = m i. P N i = l j=0 u ijn ij R : the sparse resultant of P 0,..., P n. Theorem 1 ord(r, u i ) = h i s s i, where s = n i=0 s i. 2 deg(r) n i=0 (m i + 1) h i +1 (m + 1) ns+n+1, where m = max i {m i }. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

53 Degree of Sparse Differential Resultant Laurent Diff Essential System: P 0,..., P n ord(p i ) = s i and deg(p i ) = m i. P N i = l j=0 u ijn ij R : the sparse resultant of P 0,..., P n. Theorem 1 ord(r, u i ) = h i s s i, where s = n i=0 s i. 2 deg(r) n i=0 (m i + 1) h i +1 (m + 1) ns+n+1, where m = max i {m i }. 3 n i=0 N(h i +1)deg(R) i0 R = n hi i=0 j=0 G ) ij( P N (j) i where deg(g ij (P N i ) (j) ) [m n i=0 (h i + 1)deg(N i0 )]deg(r). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

54 Key Ingredients to Prove the Theorem R [P N 0,..., PN n ] : m Q{u 0,..., u n } Differential 1) (R) = (P (k) i : k h i ) : m [h] Q[U] where h i s s i Algebraic 2) ( ) Bound of deg(r) and deg(g ik P N (k)) i Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

55 Key Ingredients to Prove the Theorem R [P N 0,..., PN n ] : m Q{u 0,..., u n } Differential 1) (R) = (P (k) i : k h i ) : m [h] Q[U] where h i s s i Algebraic 2) ( ) Bound of deg(r) and deg(g ik P N (k)) i Key facts used in 1): Diff specialization technique from diff resultant to sparse case Substitute (P N i l i k=1 u ikn ik )/N i0 for u i0 into R and Y is a parametric set. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

56 Key Ingredients to Prove the Theorem R [P N 0,..., PN n ] : m Q{u 0,..., u n } Differential 1) (R) = (P (k) i : k h i ) : m [h] Q[U] where h i s s i Algebraic 2) ( ) Bound of deg(r) and deg(g ik P N (k)) i Key facts used in 1): Diff specialization technique from diff resultant to sparse case Substitute (P N i l i k=1 u ikn ik )/N i0 for u i0 into R and Y is a parametric set. Key facts used in 2): Bézout Inequality: deg(v ) r i=1 deg(f i) for each component V of V(f 1,..., f r ). Degree of Elimination Ideal: deg(i k ) deg(i). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

57 An Algorithm for Sparse Differential Resultant Outline of the Algorithm. Knowing order and degree bounds, we compute sparse diff resultant by solving linear equations. Precisely, Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

58 An Algorithm for Sparse Differential Resultant Outline of the Algorithm. Knowing order and degree bounds, we compute sparse diff resultant by solving linear equations. Precisely, 1 Search for R(u 0,..., u n ) with order h i = 0,..., s s i and with degree from D = 1,..., n i=0 (m i + 1) h i With fixed h i and D, computing coefficients of R and G ik by solving linear equations raising from n i=0 N(h i +1)D i0 R(u 0,..., u n ) = n hi i=0 k=0 G ikp (k) i. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

59 An Algorithm for Sparse Differential Resultant Outline of the Algorithm. Knowing order and degree bounds, we compute sparse diff resultant by solving linear equations. Precisely, 1 Search for R(u 0,..., u n ) with order h i = 0,..., s s i and with degree from D = 1,..., n i=0 (m i + 1) h i With fixed h i and D, computing coefficients of R and G ik by solving linear equations raising from n i=0 N(h i +1)D i0 R(u 0,..., u n ) = n hi i=0 k=0 G ikp (k) i. Theorem (Computing Complexity) O(((n + 1)(s + 1)) O(ls) (m + 1) O(nls2) ) Q-arithmetic operations. n: number of variables; s: order of system; l: size of sparse system Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

60 Sparse diff Resultant for Diff polynomials with Non-vanishing Degree Zero Terms Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

61 Diff Essential System and its Sparse Diff Resultant Sparse Diff Polynomials: P i = u i0 + l i j=1 u ijm ij (i = 0,..., n). Diff Essential System: Laurent Diff Essential. Resultant here has additional better properties. Theorem (P i, u i ) specialize to (P i, v i ). Z 2 : set of (v 0,..., v n ) s.t. P i have a common solution. Then Z 2 = V(sat(Res)). Assume 1) Any n of the P i diff independent and 2) e j Span Z {α ij }. If R(v 0,..., v n ) = 0 and R (v 0,..., v n ) 0, then u (h i ) ik P i = 0 have a unique common solution. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

62 Theorem P 0,..., P n : Diff essential system. ord(p i ) = s i and deg(p i, Y) = m i. Then R = n hi i=0 j=0 G ijp (j) i with deg(g ij P (j) i ) (m + 1)deg(R). Better computing complexity O(n (s + 1) O(n) (m + 1) O(nls2) ). Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

63 Theorem P 0,..., P n : Diff essential system. ord(p i ) = s i and deg(p i, Y) = m i. Then R = n hi i=0 j=0 G ijp (j) i with deg(g ij P (j) i ) (m + 1)deg(R). Better computing complexity O(n (s + 1) O(n) (m + 1) O(nls2) ). Degree Bound for the Diff Resultant can be improved: Theorem Diff resultant has degree bounded by s s k +1 m k n i=0 ms s i +1 i in each u k. Key Tool: The degree of Generalized Chow form. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

64 Summary We introduce the concepts of Laurent diff polynomials and Laurent diff essential systems, and give a criterion for Laurent differentially essential system in terms of the supports. Sparse differential resultant is defined and properties similar to that of the Macaulay resultant are given. A single exponential algorithm to compute the sparse differential resultant is given. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

65 Problems for Future Research Find a determinant representation for the (sparse) differential resultant. Bound the degree of the sparse diff resultant in terms of the mixed volume of the polytopes generated by the supports of the diff polynomials. Find practically efficient algorithms to compute the sparse differential resultants. Study differential toric variety further to give a necessary and sufficient condition for the sparse differential resultant vanishing. Wei Li (AMSS, CAS) Sparse Differential Resultant / 40

66 Reference: Wei Li, Xiao-Shan Gao, Chun-Ming Yuan. Sparse Differential Resultant. In Proc. ISSAC 2011, San Jose, CA, USA, , ISSAC 2011 Distinguished Paper Award. Wei Li, Chun-Ming Yuan, Xiao-Shan Gao. Sparse Differential Resultant for Laurent Differential Polynomials. ArXiv: v1, page 1-57, 4 Nov Wei Li (AMSS, CAS) Sparse Differential Resultant / 40