Differential and Difference Chow Form, Sparse Resultant, and Toric Variety

Transcription

1 Differential and Difference Chow Form, Sparse Resultant, and Toric Variety Xiao-Shan Gao Academy of Mathematics and Systems Science Chinese Academy of Sciences Xiao-Shan Gao (AMSS, CAS) / 48

2 Outline Background Sparse Differential Resultant Differential Chow Form Difference Binomial and Toric Variety Xiao-Shan Gao (AMSS, CAS) / 48

3 Sparse Differential Resultant for Laurent Differential Polynomials Xiao-Shan Gao (AMSS, CAS) / 48

4 Sylvester Resultant Two polynomials: f = a l x l + a l 1 x l a 1 x + a 0 g = b m x m + b m 1 x m b 1 x + b 0. a l a l 1 a l 2 a 0 a l a l 1 a l 2 a Res(f, g) = a l a l 1 a l 2 a 0 b m b m 1 b m 2 b 0. b m b m 1 b m 2 b b m b m 1 b m 2 b 0 Property: Res(f, g) = 0 f (x) = g(x) = 0 has common solutions J.J. Sylvester, Phil Trans of Royal Soc of London, , Xiao-Shan Gao (AMSS, CAS) / 48

5 A Brief History of Resultant Algebraic Resultant Sylvester (1883) resultant for two polynomials (n = 1) Macaulay (1902) multivariate resultant Gelfand & Sturmfels (1994) sparse resultant Xiao-Shan Gao (AMSS, CAS) / 48

6 A Brief History of Resultant Algebraic Resultant Sylvester (1883) resultant for two polynomials (n = 1) Macaulay (1902) multivariate resultant Gelfand & Sturmfels (1994) sparse resultant Differential Resultant Ritt (1932): Differential resultant for n = 1. Ferro (1997): Diff-Res as Macaulay resultant. Not complete. Zwillinger (1998): Handbook of Differential Equations. No rigorous definition for differential multi-variate resultant No study of differential sparse resultant Xiao-Shan Gao (AMSS, CAS) / 48

7 Sparse Differential Polynomials Sparse Differential Polynomials: with fixed monomials Most differential polynomials in practice are sparse y y y y Dense Diff Polynomials f = i+j 5 y i (y ) j Sparse Diff Polynomials f = + y 4 + y 5 + y 2 y 2 Xiao-Shan Gao (AMSS, CAS) / 48

8 Notations Ordinary differential field: (F, δ), e.g. (Q(x), d dx ) Diff Indeterminates: Y = {y 1,..., y n }. Notation: y (k) i = δ k y i. Laurent Diff Monomial: M = n o (l) k=1 l=0 (y k )d kl with d kl Z; Laurent Diff Poly: f = m k=1 a km k, M k Laurent diff monomials. Support of f : A = {M 1,..., M m }. Laurent Diff Poly Ring: F{Y ± }. Example. Laurent Differential Polynomial P = y 1 + y 1 y 2 P = 1 + y 1 1 y 1 y 2 Xiao-Shan Gao (AMSS, CAS) / 48

9 Differential Dimension Conjecture in Generic Case Intersection Theorem is not true in diff case: dim(v W ) dim(v ) + dim(w ) n Xiao-Shan Gao (AMSS, CAS) / 48

10 Differential Dimension Conjecture in Generic Case Intersection Theorem is not true in diff case: dim(v W ) dim(v ) + dim(w ) n Theorem I F{Y}: a prime diff ideal with dimension d > 0 and order h. f : a generic diff poly of order s with u f the set of its coefficients. Then I 1 = [I, f ] is a prime diff ideal in F u f {Y} with dimension d 1 and order h + s. Xiao-Shan Gao (AMSS, CAS) / 48

11 Differential Dimension Conjecture in Generic Case Intersection Theorem is not true in diff case: dim(v W ) dim(v ) + dim(w ) n Theorem I F{Y}: a prime diff ideal with dimension d > 0 and order h. f : a generic diff poly of order s with u f the set of its coefficients. Then I 1 = [I, f ] is a prime diff ideal in F u f {Y} with dimension d 1 and order h + s. Dimension Conjecture (Ritt, 1950): dim[f 1,..., f r ] n r. Xiao-Shan Gao (AMSS, CAS) / 48

12 Differential Dimension Conjecture in Generic Case Intersection Theorem is not true in diff case: dim(v W ) dim(v ) + dim(w ) n Theorem I F{Y}: a prime diff ideal with dimension d > 0 and order h. f : a generic diff poly of order s with u f the set of its coefficients. Then I 1 = [I, f ] is a prime diff ideal in F u f {Y} with dimension d 1 and order h + s. Dimension Conjecture (Ritt, 1950): dim[f 1,..., f r ] n r. Theorem (Generic Dimension Theorem) f 1,..., f r (r n): generic diff polynomials. Then [f 1,..., f r ]: a prime diff ideal of dimension n r and order i ord(f i). Xiao-Shan Gao (AMSS, CAS) / 48

13 Sparse Differential Resultant Generic Sparse Differential Polynomials: A i = {M i0, M i1,..., M ili } (i = 0,..., n): Monomial sets P i = l i j=0 u ijm ij and u i = {u i1,..., u ili }. [P 0, P 1,..., P n ] Q{u 0, u 1,..., u n, Y, Y 1 } Xiao-Shan Gao (AMSS, CAS) / 48

14 Sparse Differential Resultant Generic Sparse Differential Polynomials: A i = {M i0, M i1,..., M ili } (i = 0,..., n): Monomial sets P i = l i j=0 u ijm ij and u i = {u i1,..., u ili }. [P 0, P 1,..., P n ] Q{u 0, u 1,..., u n, Y, Y 1 } Sparse Differential Resultant Exists, if the eliminant ideal: [P 0,..., P n ] Q{u 0, u 1,..., u n } = sat(r(u 0,..., u n )) is of codimension 1 Definition R: Sparse Differential Resultant of P 0,..., P n or A 0,..., A n. Xiao-Shan Gao (AMSS, CAS) / 48

15 Sparse Differential Resultant Generic Sparse Differential Polynomials: A i = {M i0, M i1,..., M ili } (i = 0,..., n): Monomial sets P i = l i j=0 u ijm ij and u i = {u i1,..., u ili }. [P 0, P 1,..., P n ] Q{u 0, u 1,..., u n, Y, Y 1 } Sparse Differential Resultant Exists, if the eliminant ideal: [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: Definition 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. R: Sparse Differential Resultant of P 0,..., P n or A 0,..., A n. Xiao-Shan Gao (AMSS, CAS) / 48

16 Examples Example n = 2, P i = u i0 y 1 + u i1y 1 + u i2y 2 d.tr.deg Q y 1 y 1 (i = 0, 1, 2)., y 2 /Q = 2 = P y 1 i form a diff essential system. The sparse differential resultant is u 00 u 01 u 02 R = u 10 u 11 u 12 u 20 u 21 u 22. Xiao-Shan Gao (AMSS, CAS) / 48

17 Criterion for Existence of Sparse Resultant P i = l i j=0 u ijm ij (i = 0,..., n). si (l) l=0 (y k )d ijkl. d ijk = s i l=0 d ijklxk l Q[x k]. M ij /M i0 = n k=1 Symbolic Support Vector of M ij /M i0 : β ij = (d ij1,..., d ijn ) Xiao-Shan Gao (AMSS, CAS) / 48

18 Criterion for Existence of Sparse Resultant P i = l i j=0 u ijm ij (i = 0,..., n). si (l) l=0 (y k )d ijkl. d ijk = s i l=0 d ijklxk l Q[x k]. M ij /M i0 = n k=1 Symbolic Support Vector of M ij /M i0 : β ij = (d ij1,..., d ijn ) Symbolic Support Vector of P i : β i = M P = l i j=0 Symbolic Support Matrix of P 0,..., P n : β 0 β 1. β n u ij β ij = (d i1,..., d in ). d 01 d d 0n = d 11 d d 1n... d n1 d n2... d nn Xiao-Shan Gao (AMSS, CAS) / 48

19 Criterion for Existence of Sparse Resultant P i = l i j=0 u ijm ij (i = 0,..., n). si (l) l=0 (y k )d ijkl. d ijk = s i l=0 d ijklxk l Q[x k]. M ij /M i0 = n k=1 Symbolic Support Vector of M ij /M i0 : β ij = (d ij1,..., d ijn ) Symbolic Support Vector of P i : β i = M P = l i j=0 Symbolic Support Matrix of P 0,..., P n : β 0 β 1. β n u ij β ij = (d i1,..., d in ). d 01 d d 0n = d 11 d d 1n... d n1 d n2... d nn Theorem (Like Linear Algebra!) Sparse resultant exists for P i rk(m P ) = n. Xiao-Shan Gao (AMSS, CAS) / 48

20 Properties of Sparse Differential Resultant Xiao-Shan Gao (AMSS, CAS) / 48

21 Necessary Condition for of Non-poly Solutions Lemma (P i, u i ) specializes to (P i, v i ) by setting u i = v i F. If P 0 = = P n = 0 has a non-poly solution, then R(v 0,..., v n ) = 0. Xiao-Shan Gao (AMSS, CAS) / 48

22 Necessary Condition for of Non-poly Solutions Lemma (P i, u i ) specializes to (P i, v i ) by setting u i = v i F. If P 0 = = P n = 0 has a non-poly solution, then R(v 0,..., v n ) = 0. Example (Why Non-Polynomial solution?) F = Q(x), differential operator: x P i = u i0 y 1 + u i1y 1 + u i2y 2 (i = 0, 1, 2). u 00 u 01 u 02 The sparse differential resultant R = u 10 u 11 u 12 u 20 u 21 u Let a 1 = x + 1, a 2 = x 2 + x + 1. Then a 1 = a 2 = 0. (a 1, a 2 ): a solution of P 0 = P 1 = P 2 = 0 Xiao-Shan Gao (AMSS, CAS) / 48

23 Conditions for Existence of Non-poly Solutions A i = (M i0,..., M ili ) : Differential Monomials Xiao-Shan Gao (AMSS, CAS) / 48

24 Conditions for Existence of Non-poly Solutions A i = (M i0,..., M ili ) : Differential Monomials L(A i ) = {F i = l i j=0 c im ij }: all diff polys with support A i. Xiao-Shan Gao (AMSS, CAS) / 48

25 Conditions for Existence of Non-poly Solutions A i = (M i0,..., M ili ) : Differential Monomials L(A i ) = {F i = l i j=0 c im ij }: all diff polys with support A i. Z 0 (A 0,..., A n ): set of F i having a common non-poly solution. Z 0 (A 0,..., A n ): Kolchin diff closure of Z 0 (A 0,..., A n ). Xiao-Shan Gao (AMSS, CAS) / 48

26 Conditions for Existence of Non-poly Solutions A i = (M i0,..., M ili ) : Differential Monomials L(A i ) = {F i = l i j=0 c im ij }: all diff polys with support A i. Z 0 (A 0,..., A n ): set of F i having a common non-poly solution. Z 0 (A 0,..., A n ): Kolchin diff closure of Z 0 (A 0,..., A n ). Theorem Z 0 (A 0,..., A n ) = V ( sat(res A0,...,A n ) ). On a Kolchin open set of V ( sat(res A0,...,A n ) ), F 0 = = F n = 0 have non-poly solutions Res F0,...,F n = 0. Xiao-Shan Gao (AMSS, CAS) / 48

27 Order and Differential homogeneity G = {g 1,..., g n }: differential polynomials. Jacobi Number: Jac(G) = max σ n i=1 ord(g i, y σ (i)), where σ is a permutation of {1,..., n}. Xiao-Shan Gao (AMSS, CAS) / 48

28 Order and Differential homogeneity G = {g 1,..., g n }: differential polynomials. Jacobi Number: Jac(G) = max σ n i=1 ord(g i, y σ (i)), where σ is a permutation of {1,..., n}. Order and Differential homogeneity δres(u 0,..., u n ) is differentially 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. S-δRes(u 0,..., u n ) is differentially homogeneous in each u i and is of order h i J i = Jac(Pî) in u i, where Pî = {P 0,..., P n }\{P i }. Xiao-Shan Gao (AMSS, CAS) / 48

29 Poisson-Type Product Formula Algebraic Resultant: Res(A(x), B(x)) = c η,b(η)=0 A(η). Xiao-Shan Gao (AMSS, CAS) / 48

30 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 ]. Xiao-Shan Gao (AMSS, CAS) / 48

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 ]. Sparse Differential Resultant: S-δRes(u 0,..., u n ) = A t 0 τ=1 (u 00 + l 0k=1 u 0k ξ τk ) (h 0). Xiao-Shan Gao (AMSS, CAS) / 48

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: S-δRes(u 0,..., u n ) = A t 0 τ=1 (u 00 + l 0k=1 u 0k ξ τk ) (h 0). 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: S-δRes(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 ) are generic points of [P N 0,..., PN n ] : m. Xiao-Shan Gao (AMSS, CAS) / 48

33 Poisson-Type Product Formula Sylvester Resultant: Res(A(x), B(x)) = c η,b(η)=0 A(η). Xiao-Shan Gao (AMSS, CAS) / 48

34 Poisson-Type Product Formula Sylvester 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 ]. Xiao-Shan Gao (AMSS, CAS) / 48

35 Poisson-Type Product Formula Sylvester 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 ]. Sylvester Resultant: Res(A(x), B(x)) = A(x)T(x) + B(x)W(x), where deg(t ) < deg(b), deg(w ) < deg(a). Xiao-Shan Gao (AMSS, CAS) / 48

36 Poisson-Type Product Formula Sylvester 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 ]. Sylvester 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 where s i = ord(p i ) and s = s s n, and deg(g ij P (j) i ) (m + 1)deg(R) (m + 1) ns+n+2. Xiao-Shan Gao (AMSS, CAS) / 48

37 Degree Bound of Sparse Differential Resultant Laurent Diff Essential System: P i, ord(p i ) = s i and deg(p i ) = m i. R : the sparse resultant of P 0,..., P n. Xiao-Shan Gao (AMSS, CAS) / 48

38 Degree Bound of Sparse Differential Resultant Laurent Diff Essential System: P i, ord(p i ) = s i and deg(p i ) = m i. R : the sparse resultant of P 0,..., P n. Theorem (Degree Bounds) 1 deg(r) n i=0 (m i + 1) h i +1 (m + 1) ns+n+1, where m = max i {m i }. Xiao-Shan Gao (AMSS, CAS) / 48

39 Degree Bound of Sparse Differential Resultant Laurent Diff Essential System: P i, ord(p i ) = s i and deg(p i ) = m i. R : the sparse resultant of P 0,..., P n. Theorem (Degree Bounds) 1 deg(r) n i=0 (m i + 1) h i +1 (m + 1) ns+n+1, where m = max i {m i }. 2 R = n s si i=0 j=0 h ijp (j) i deg(g ij P (j) i ) (m + 1)deg(R) (m + 1) ns+n+2. Xiao-Shan Gao (AMSS, CAS) / 48

40 BKK Degree Bound for Differential Resultant Theorem P i (i = 0,..., n): generic diff polynomials in Y with order s i, coefficient set u i, and s = n i=0 s i. Then deg(r, u i ) s s i k=0 M( (Q jl ) j i,0 l s sj, Q i0,..., Q i,k 1, Q i,k+1,..., Q i,s si ). Q jl : Newton polytope of P (l) j as a polynomial in y [s] 1,..., y [s] n. M(Q 1,..., Q n ): Mixed volume of Q 1,..., Q n. Xiao-Shan Gao (AMSS, CAS) / 48

41 BKK Degree Bound for Differential Resultant Theorem P i (i = 0,..., n): generic diff polynomials in Y with order s i, coefficient set u i, and s = n i=0 s i. Then deg(r, u i ) s s i k=0 M( (Q jl ) j i,0 l s sj, Q i0,..., Q i,k 1, Q i,k+1,..., Q i,s si ). Q jl : Newton polytope of P (l) j as a polynomial in y [s] 1,..., y [s] n. M(Q 1,..., Q n ): Mixed volume of Q 1,..., Q n. Example P 0 = u 00 + u 01 y + u 02 y + u 03 y 2 + u 04 yy + u 05 (y ) 2 P 1 = u 10 + u 11 y + u 12 y + u 13 y 2 + u 14 yy + u 15 (y ) 2 Bézout-type degree bound: deg(r) (2 + 1) 4 = 81. BKK-type degree bound: deg(r) 20. Xiao-Shan Gao (AMSS, CAS) / 48

42 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, Xiao-Shan Gao (AMSS, CAS) / 48

43 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 R(u 0,..., u n ) = n hi i=0 k=0 h ikp (k) i. Xiao-Shan Gao (AMSS, CAS) / 48

44 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 R(u 0,..., u n ) = n hi i=0 k=0 h ikp (k) i. Theorem (Computing Complexity) O(m O(nls2) ) Q-arithmetic operations. n: number of variables; s: order of system; l: size of sparse system Xiao-Shan Gao (AMSS, CAS) / 48

45 Difference Sparse Resultant Comparison with differential sparse resultant: Difference Case Differential Case Definition sat(r, R 1,..., R m ) sat(r) Problem: m = 0? Criterion M P Z[x] (n+1) n M P Z[u ij, x 1,..., x n ] (n+1) n Matrix R = det(m)/ det(m 0 )? solutions Necessary non-zero sols Nec and Suff non-poly sol Z 0 = V(sat(R, R 1,..., R m )) Z 0 = V(sat(R)) Homogeneity Transformally homogenous Differentially homogenous f (λy) = M(λ)f (Y) f (λy) = λ m f (Y) Degree Dense: = BKK number Dense: BKK bound Sparse: Bezout Type bound Sparse: Bezout Type bound Order Sparse: Jacobi bound The same Dense: s s i The same Xiao-Shan Gao (AMSS, CAS) / 48

46 Differential Chow Form Xiao-Shan Gao (AMSS, CAS) / 48

47 Example: Plücker Coordinates Using coordinates to represent algebraic variety Xiao-Shan Gao (AMSS, CAS) / 48

48 Example: Plücker Coordinates Using coordinates to represent algebraic variety Lines in P(3): { a0 x Line L := 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 = 0 b 0 x 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 = 0 (one to one correspondence) Plücker Coordinates: p ij = a i a j, i, j = 0, 1, 2, 3 b i b j Xiao-Shan Gao (AMSS, CAS) / 48

49 Example: Plücker Coordinates Using coordinates to represent algebraic variety Lines in P(3): { a0 x Line L := 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 = 0 b 0 x 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 = 0 (one to one correspondence) Plücker Coordinates: p ij = a i a j, i, j = 0, 1, 2, 3 Plücker coordinate C = (p 01, p 02, p 03, p 23, p 31, p 12 ) P(5) C represents a line in P(3) C is on hypersurface p 23 p 01 + p 31 p 02 + p 12 p 03 = 0. b i b j Xiao-Shan Gao (AMSS, CAS) / 48

50 Background: Chow Form Using coordinates to represent algebraic varietyµ Lines in P(3): Plücker Coordinates {Line L P(3)} hypersurface p 23 p 01 + p 31 p 02 + p 12 p 03 = 0. Xiao-Shan Gao (AMSS, CAS) / 48

51 Background: Chow Form Using coordinates to represent algebraic varietyµ Lines in P(3): Plücker Coordinates {Line L P(3)} hypersurface p 23 p 01 + p 31 p 02 + p 12 p 03 = 0. Subspace of d-dim in P(n)µ Grassmann Coordinates {S d P(n)} Grassmann Variety Xiao-Shan Gao (AMSS, CAS) / 48

52 Background: Chow Form Using coordinates to represent algebraic varietyµ Lines in P(3): Plücker Coordinates {Line L P(3)} hypersurface p 23 p 01 + p 31 p 02 + p 12 p 03 = 0. Subspace of d-dim in P(n)µ Grassmann Coordinates {S d P(n)} Grassmann Variety Algebraic Variety in P(n): Chow Coordinates {(r,d) cycles} Chow Variety Xiao-Shan Gao (AMSS, CAS) / 48

53 Background: Chow Form Using coordinates to represent algebraic varietyµ Lines in P(3): Plücker Coordinates {Line L P(3)} hypersurface p 23 p 01 + p 31 p 02 + p 12 p 03 = 0. Subspace of d-dim in P(n)µ Grassmann Coordinates {S d P(n)} Grassmann Variety Algebraic Variety in P(n): Chow Coordinates {(r,d) cycles} Chow Variety Differential Analog? Xiao-Shan Gao (AMSS, CAS) / 48

54 Definition of Differential Chow Form I F{Y}: prime differential ideal of dimension d. d + 1 Generic Differential Primes: P i = u i0 + u i1 y u in y n (i = 0,..., d). u i = (u i0,..., u in ): coefficient set of P i Xiao-Shan Gao (AMSS, CAS) / 48

55 Definition of Differential Chow Form I F{Y}: prime differential ideal of dimension d. d + 1 Generic Differential Primes: P i = u i0 + u i1 y u in y n (i = 0,..., d). u i = (u i0,..., u in ): coefficient set of P i Theorem By intersecting I with the d + 1 primes, the eliminant ideal [I, P 0,..., P d ] F{u 0, u 1,..., u d } = sat(f(u 0, u 1,..., u d )) is a prime ideal of co-dimension one. Differential Chow form of I or V(I): F(u 0, u 1,..., u d ) = f (u; u 00,..., u d0 ) Xiao-Shan Gao (AMSS, CAS) / 48

56 Order of Differential Chow Form Chow form of I: F(u 0, u 1,..., u d ) = f (u; u 00, u 10,..., u d0 ) Property of Chow form. F(..., u σ,..., u ρ,...) = ( 1) rσρ F(..., u ρ,..., u σ,...). ord(f, u 00 ) 0, ord(f, u 00 ) = ord(f, u ij ) if u ij occurs in F Xiao-Shan Gao (AMSS, CAS) / 48

57 Order of Differential Chow Form Chow form of I: F(u 0, u 1,..., u d ) = f (u; u 00, u 10,..., u d0 ) Property of Chow form. F(..., u σ,..., u ρ,...) = ( 1) rσρ F(..., u ρ,..., u σ,...). ord(f, u 00 ) 0, ord(f, u 00 ) = ord(f, u ij ) if u ij occurs in F Order of Chow form: ord(f) = ord(f, u 00 ). Xiao-Shan Gao (AMSS, CAS) / 48

58 Order of Differential Chow Form Chow form of I: F(u 0, u 1,..., u d ) = f (u; u 00, u 10,..., u d0 ) Property of Chow form. F(..., u σ,..., u ρ,...) = ( 1) rσρ F(..., u ρ,..., u σ,...). ord(f, u 00 ) 0, ord(f, u 00 ) = ord(f, u ij ) if u ij occurs in F Order of Chow form: ord(f) = ord(f, u 00 ). Theorem (Order of Chow Form) ord(f) = ord(i). Xiao-Shan Gao (AMSS, CAS) / 48

59 Degree of Differential Chow Form Differentially homogenous diff poly of degree m: p(ty 0, ty 1..., ty n ) = t m p(y 0, y 1,..., y n ) Theorem F(u 0, u 1,..., u d ): differential Chow form of V. Then F (u 0, u 1,..., u d ) is differentially homogenous of degree r in each set u i and F is of total degree (d + 1)r. Definition (Differential degree) r as above is defined to be the differential degree of I, which is an invariant of I under invertible linear transformations. Xiao-Shan Gao (AMSS, CAS) / 48

60 Factorization of Differential Chow Form V : a diff irreducible variety of dimension d and order h. F(u 0, u 1,..., u d ): the differential Chow form of V. Theorem (F(u 0, u 1,..., u d ) can be uniquely factored) F(u 0, u 1,..., u d ) = A(u 0, u 1,..., u d ) = A(u 0, u 1,..., u d ) g (u 00 + τ=1 n u 0ρ ξ τρ ) (h) ρ=1 g P 0 (ξ τ1,..., ξ τn ) (h) where g = deg(f, u (h) 00 ) and ξ τρ are in an extension field of F. And the points (ξ τ1,..., ξ τn ) (τ = 1,..., g) are generic points of the variety V. τ=1 Xiao-Shan Gao (AMSS, CAS) / 48

61 Leading Differential Degree Differential primes: P i := u i0 + u i1 y u in y n (i = 1,..., d), Algebraic primes: a P 0 := u 00 + u 01 y u 0n y n, a P (s) 0 := u (s) 00 + n s ( s ) (k) j=1 k=0 k u 0j y (s k) j (s = 1, 2,...) Theorem (ξ τ1,..., ξ τn ) (τ = 1,..., g) are the only elements of V which lie on P 1,..., P d as well as on a P 0, a P 0,...,a P (h 1) 0. Definition Number g is defined to be the leading diff degree of V or I. Xiao-Shan Gao (AMSS, CAS) / 48

62 Differential Chow variety A diff variety V has index (n, d, h, g, m) if V E n has invariants: dim d, order h, leading diff degree g, and diff degree m. Xiao-Shan Gao (AMSS, CAS) / 48

63 Differential Chow variety A diff variety V has index (n, d, h, g, m) if V E n has invariants: dim d, order h, leading diff degree g, and diff degree m. Diff Cycle: V = i s iv i, V i irreducible of index (d, h, g, m) Chow Form of V : i F s i i, F i Chow form of V i Index of V : (d, h, i s ig i, i s im i ) Xiao-Shan Gao (AMSS, CAS) / 48

64 Differential Chow variety A diff variety V has index (n, d, h, g, m) if V E n has invariants: dim d, order h, leading diff degree g, and diff degree m. Diff Cycle: V = i s iv i, V i irreducible of index (d, h, g, m) Chow Form of V : i F s i i, F i Chow form of V i Index of V : (d, h, i s ig i, i s im i ) Definition A diff variety V is a Chow Variety if (ā i ) V F with coef (ā i ): Chow form with index (n, d, h, g, m). V : diff cycle of index (n, d, h, g, m). Xiao-Shan Gao (AMSS, CAS) / 48

65 Differential Chow variety A diff variety V has index (n, d, h, g, m) if V E n has invariants: dim d, order h, leading diff degree g, and diff degree m. Diff Cycle: V = i s iv i, V i irreducible of index (d, h, g, m) Chow Form of V : i F s i i, F i Chow form of V i Index of V : (d, h, i s ig i, i s im i ) Definition A diff variety V is a Chow Variety if (ā i ) V F with coef (ā i ): Chow form with index (n, d, h, g, m). V : diff cycle of index (n, d, h, g, m). Chow Coordinate of V : (ā i ) In affine case, Chow Variety is a constructible set. Xiao-Shan Gao (AMSS, CAS) / 48

66 Differential Chow variety Theorem (Gao-Li-Yuan, 2013) In the case g = 1, the differential Chow variety exists. Difficulty for the general case: Eliminating both differential and algebraic variables. Xiao-Shan Gao (AMSS, CAS) / 48

67 Differential Chow variety Theorem (Gao-Li-Yuan, 2013) In the case g = 1, the differential Chow variety exists. Difficulty for the general case: Eliminating both differential and algebraic variables. Theorem (Freitag-Li-Scanlon, 2015) The differential Chow variety exists. Key Ideas: Use prolongation admissible varieties and prolongation sequences to reduce the construction to the algebraic case. Definability in ACF and DCF 0 is also used. Xiao-Shan Gao (AMSS, CAS) / 48

68 Differential Toric Variety Xiao-Shan Gao (AMSS, CAS) / 48

69 Differential Toric Variety A = {M 0,..., M l }: a set of diff monomials Xiao-Shan Gao (AMSS, CAS) / 48

70 Differential Toric Variety A = {M 0,..., M l }: a set of diff monomials Consider the map φ A : (E ) n P(l) η (M 0 (η), M 1 (η),..., M l (η)) Xiao-Shan Gao (AMSS, CAS) / 48

71 Differential Toric Variety A = {M 0,..., M l }: a set of diff monomials Consider the map φ A : (E ) n P(l) η (M 0 (η), M 1 (η),..., M l (η)) Definition The image of φ A is called the differential toric variety w.r.t. A, denoted by X A. X A is an irreducible projective diff variety. Xiao-Shan Gao (AMSS, CAS) / 48

72 Differential Toric Variety A = {M 0,..., M l }: a set of diff monomials Consider the map φ A : (E ) n P(l) η (M 0 (η), M 1 (η),..., M l (η)) Definition The image of φ A is called the differential toric variety w.r.t. A, denoted by X A. X A is an irreducible projective diff variety. Theorem Res A : Sparse differential resultant of P i = j u ijm j, i = 0,..., n is the differential Chow form of X A. Xiao-Shan Gao (AMSS, CAS) / 48

73 Differential Toric Variety: An Example Example Let n = 1 and a set of monomials A = {y 1, y 1, y 2 1 }. Toric Variety: all possible values of a set of monomials A X A = {(y 1, y 1, y 2 1 ) y 1 E} Defining equations of the toric variety X A = Zero(sat(z 1 z 2 (z 0 z 2 z 0 z 2))). The sparse differential resultant Res A is equal to the differential Chow form of X A. Xiao-Shan Gao (AMSS, CAS) / 48

74 Differential Toric Variety: An Example Example Let n = 1 and a set of monomials A = {y 1, y 1, y 2 1 }. Toric Variety: all possible values of a set of monomials A X A = {(y 1, y 1, y 2 1 ) y 1 E} Defining equations of the toric variety X A = Zero(sat(z 1 z 2 (z 0 z 2 z 0 z 2))). The sparse differential resultant Res A is equal to the differential Chow form of X A. The defining ideal of a diff toric variety is not binomial! Xiao-Shan Gao (AMSS, CAS) / 48

75 Binomial σ-ideal and Toric σ-variety Xiao-Shan Gao (AMSS, CAS) / 48

76 Notations In this talk, difference field (F, σ): σ : F F is a field automorphism. F is also assumed to be algebraically closed. Example: Q(x) : σ(x) = x + 1. Xiao-Shan Gao (AMSS, CAS) / 48

77 Notations In this talk, difference field (F, σ): σ : F F is a field automorphism. F is also assumed to be algebraically closed. Example: Q(x) : σ(x) = x + 1. σ-exponent: For p = s i=0 c ix i Z[x], denote a p = s i=0 (σi a) c i. Example: a 3x 2 1 = (σ 2 (a)) 3 /a Xiao-Shan Gao (AMSS, CAS) / 48

78 Notations In this talk, difference field (F, σ): σ : F F is a field automorphism. F is also assumed to be algebraically closed. Example: Q(x) : σ(x) = x + 1. σ-exponent: For p = s i=0 c ix i Z[x], denote a p = s i=0 (σi a) c i. Example: a 3x 2 1 = (σ 2 (a)) 3 /a Y = {y 1,..., y n }: σ-indeterminates σ-monomial with support f: Y f = n i=1 y f i i where f = (f 1,..., f n ) τ N[x] n F{Y}: σ-polynomial ring Xiao-Shan Gao (AMSS, CAS) / 48

79 Z[x] Lattice Z[x] Lattice: Z[x] module in Z[x] n Two kinds of representations: Generators: L = Span Z[x] {f 1,..., f s } = (f 1,..., f s ), f i Z[x] n Matrix representation: F = [f 1,..., f s ] n s Rank of L: rk(f) Xiao-Shan Gao (AMSS, CAS) / 48

80 Binomial σ-ideal σ-binomial: f = ay a + by b, a, b N[x] n, a, b F. Normal Form: f = ay g (Y f+ cy f ), f Z[x] n and f = a b = f + f for f +, f N[x] n. Xiao-Shan Gao (AMSS, CAS) / 48

81 Binomial σ-ideal σ-binomial: f = ay a + by b, a, b N[x] n, a, b F. Normal Form: f = ay g (Y f+ cy f ), f Z[x] n and f = a b = f + f for f +, f N[x] n. Normal binomial σ-ideal I: I is generated by σ-binomials Mp I p I, M: σ-monomial Xiao-Shan Gao (AMSS, CAS) / 48

82 Binomial σ-ideal σ-binomial: f = ay a + by b, a, b N[x] n, a, b F. Normal Form: f = ay g (Y f+ cy f ), f Z[x] n and f = a b = f + f for f +, f N[x] n. Normal binomial σ-ideal I: I is generated by σ-binomials Mp I p I, M: σ-monomial Partial Character: A homomorphism from a Z[x] lattice L ρ to the multiplicative group F satisfying ρ(xf) = σ(ρ(f)). Lemma I is a normal binomial σ-ideal I = I(ρ) = {Y f+ ρ(f)y f f L ρ } for a partial character ρ. Xiao-Shan Gao (AMSS, CAS) / 48

83 Criteria for Normal LBσ-ideal Definition A Z[x] lattice L in Z[x] n is called Z-saturated if, for a Z and f Z[x] n, af L implies f L. x-saturated if, for f Z[x] n, xf L implies f L. M-saturated if, for f Z[x] n and m N, mf L (x o m )f L. Xiao-Shan Gao (AMSS, CAS) / 48

84 Criteria for Normal LBσ-ideal Definition A Z[x] lattice L in Z[x] n is called Z-saturated if, for a Z and f Z[x] n, af L implies f L. x-saturated if, for f Z[x] n, xf L implies f L. M-saturated if, for f Z[x] n and m N, mf L (x o m )f L. Theorem Let ρ be a partial character over Z[x] n. L ρ is Z-saturated I(ρ) is prime L ρ is x-saturated I(ρ) is reflexive If I(ρ) : M [1], then L ρ is M-saturated I(ρ) is well-mixed If {I(ρ)} : M [1], then L ρ is x-m-saturated I(ρ) is perfect Xiao-Shan Gao (AMSS, CAS) / 48

85 Toric σ-variety For α = {α 1,..., α n }, α i Z[x] m, i = 1,..., n Define a map φ α : (A ) m (A ) n : T = (t 1,..., t m ) T α = (T α 1,..., T αn ). Xiao-Shan Gao (AMSS, CAS) / 48

86 Toric σ-variety For α = {α 1,..., α n }, α i Z[x] m, i = 1,..., n Define a map φ α : (A ) m (A ) n : T = (t 1,..., t m ) T α = (T α 1,..., T αn ). Toric Variety X α : the Cohn closure of φ α ((C ) m ) in (A) n. Toric Variety: σ-variety parameterized by σ-monomials. X α is an irreducible σ-variety of dim rk(a), where A = [α 1,..., α n ] m n Xiao-Shan Gao (AMSS, CAS) / 48

87 Toric σ-variety For α = {α 1,..., α n }, α i Z[x] m, i = 1,..., n Define a map φ α : (A ) m (A ) n : T = (t 1,..., t m ) T α = (T α 1,..., T αn ). Toric Variety X α : the Cohn closure of φ α ((C ) m ) in (A) n. Toric Variety: σ-variety parameterized by σ-monomials. X α is an irreducible σ-variety of dim rk(a), where A = [α 1,..., α n ] m n Example The support: α = {[1, 1] τ, [x, x] τ, [0, 1] τ }. The σ-monomial: (t 1 t 2, t x 1 tx 2, t 2). The map: y 1 = t 1 t 2, y 2 = t x 1 tx 2, y 3 = t 2 Toric σ-variety: X α : y x 1 y 2 = 0. Note that y 3 is free. Xiao-Shan Gao (AMSS, CAS) / 48

88 Toric σ-ideal Toric Z[x] Lattice L: pf L f L (p Z[x] and f Z[x] n ) Xiao-Shan Gao (AMSS, CAS) / 48

89 Toric σ-ideal Toric Z[x] Lattice L: pf L f L (p Z[x] and f Z[x] n ) Toric σ-ideal: I + (L) = [Y f+ Y f f L], where L is a toric Z[x] lattice. I + (L) is reflexive and prime σ-ideal of dimension rk(l). Xiao-Shan Gao (AMSS, CAS) / 48

90 Toric σ-ideal Toric Z[x] Lattice L: pf L f L (p Z[x] and f Z[x] n ) Toric σ-ideal: I + (L) = [Y f+ Y f f L], where L is a toric Z[x] lattice. I + (L) is reflexive and prime σ-ideal of dimension rk(l). Theorem A σ-variety V is toric iff I(V ) is a toric σ-ideal. Xiao-Shan Gao (AMSS, CAS) / 48

91 Toric σ-ideal Toric Z[x] Lattice L: pf L f L (p Z[x] and f Z[x] n ) Toric σ-ideal: I + (L) = [Y f+ Y f f L], where L is a toric Z[x] lattice. I + (L) is reflexive and prime σ-ideal of dimension rk(l). Theorem A σ-variety V is toric iff I(V ) is a toric σ-ideal. Example (Reflexive prime but not toric) Let L = ([1 x, x 1] τ ). Since [1 x, x 1] = (x 1) [1, 1], L is not Z[x] toric. The σ-ideal I + (L) = [y x i but not toric. 1 y x j 2 y x j 1 y x i 2 ; 0 i j N] is reflexive prime Xiao-Shan Gao (AMSS, CAS) / 48

92 Conversion between V = X α and I(V ) = I + (ρ L ) (1) Implicitization: Given X α (α = (α 1,..., α n )) I(V ) F{Y} A = [α 1,..., α n ] m n K A = ker(a) = (f 1,... f s ): a toric Z[x] lattice; Gröbner basis I(X α ) = I + (K A ) = sat(y f+ 1 Y f 1,..., Y f+ s Y f s ) Xiao-Shan Gao (AMSS, CAS) / 48

93 Conversion between V = X α and I(V ) = I + (ρ L ) (1) Implicitization: Given X α (α = (α 1,..., α n )) I(V ) F{Y} A = [α 1,..., α n ] m n K A = ker(a) = (f 1,... f s ): a toric Z[x] lattice; Gröbner basis I(X α ) = I + (K A ) = sat(y f+ 1 Y f 1,..., Y f+ s Y f s ) (2) Parametrization: Given I = sat(y f+ 1 Y f 1,..., Y f+ s Y f s ) Xα = V(I) L ρ = (f 1,..., f s ) F = [f 1,..., f s ] n s Z[x] n s K F = {X Z[x] n F τ X = 0} is a free Z[x] module. K F has a basis {h 1,..., h n r } H = [h 1,..., h n r ] n (n r) α = {α 1,..., α n } Z[x] n r the rows of H. If L ρ is a toric Z[x] lattice, then X α = V(I) Xiao-Shan Gao (AMSS, CAS) / 48

94 Coordinate Ring of Toric σ-variety Affine N[x] module: β = {β 1,..., β s } Z[x] m M = N[x](β) = { s i=1 a iβ i a i N[x]} Z[x] m. Affine σ-algebra F{M} = { f M a f T f a f F, a β 0 for finitely many β}. Xiao-Shan Gao (AMSS, CAS) / 48

95 Coordinate Ring of Toric σ-variety Affine N[x] module: β = {β 1,..., β s } Z[x] m M = N[x](β) = { s i=1 a iβ i a i N[x]} Z[x] m. Affine σ-algebra F{M} = { f M a f T f a f F, a β 0 for finitely many β}. Theorem. X is a toric σ-variety X = Spec σ (Q{M}) for an affine N[x] module M. the coordinate ring of X is Q{M}. Xiao-Shan Gao (AMSS, CAS) / 48

96 Toric σ-variety in terms of group action The map φ α : (A ) m (A ) n : Quasi σ-torus: T α = φ α ((A ) m ) In the algebraic case, T α (the torus) is a variety: T α = X α (C ) m This is not valid in the difference case. Xiao-Shan Gao (AMSS, CAS) / 48

97 Toric σ-variety in terms of group action The map φ α : (A ) m (A ) n : Quasi σ-torus: T α = φ α ((A ) m ) In the algebraic case, T α (the torus) is a variety: T α = X α (C ) m This is not valid in the difference case. σ-torus: a σ-variety isomorphic to the Cohn -closure of T α in (A ) n. (1) T is a σ-variety which is open in X α. (2) A σ-torus is group under componentwise product. Xiao-Shan Gao (AMSS, CAS) / 48

98 Toric σ-variety in terms of group action The map φ α : (A ) m (A ) n : Quasi σ-torus: T α = φ α ((A ) m ) In the algebraic case, T α (the torus) is a variety: T α = X α (C ) m This is not valid in the difference case. σ-torus: a σ-variety isomorphic to the Cohn -closure of T α in (A ) n. (1) T is a σ-variety which is open in X α. (2) A σ-torus is group under componentwise product. Theorem (Toric σ-variety in terms of group action) A σ-variety X is toric iff X contains a σ-torus T as an open subset and with a group action of T on X extending the natural group action of T on itself. Xiao-Shan Gao (AMSS, CAS) / 48

99 Summary Sparse differential/difference resultant is defined and properties similar to that of the Sylvester resultant are given. A single exponential algorithm to compute the sparse differential resultant is given. Differential/difference Chow Form is defined and its basic properties are established. Difference binomial ideals and difference toric varieties are introduced, which connects the difference Chow form and difference sparse resultant. Xiao-Shan Gao (AMSS, CAS) / 48

100 Thanks! Xiao-Shan Gao (AMSS, CAS) / 48

101 Summary W. Li, C.M. Yuan, X.S. Gao. Sparse Differential Resultant for Laurent Differential Polynomials. Found of Comput Math, 15(2), , W. Li, C.M. Yuan, X.S. Gao. Sparse Difference Resultant. Journal of Symbolic Computation, 68, , X.S. Gao, W. Li, C.M. Yuan. Intersection Theory in Differential Algebraic Geometry: Generic Intersections and the Differential Chow Form. Trans. of Amer. Math. Soc., 365(9), , W. Li and Y.H. Li. Difference Chow form Journal of Algebra, 428(15), 67-90, X.S. Gao, Z. Huang, C.M. Yuan. Binomial Difference Ideal and Toric Difference Variety. arxiv: , Xiao-Shan Gao (AMSS, CAS) / 48