. Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix An algorithm to compute the differential equations for the logarithm of a polynomial . Toshinori Oaku . Tokyo Woman's Christian University .. ISSAC 2012, Grenoble, July 2012 . Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Weyl algebra Let Dn = Chx;@x i be the Weyl algebra, or the ring of differential operators with polynomial coefficients. An element P of Dn is expressed as a finite sum X α β P = aα,βx @ α,β2Nn α α1 ··· αn β β1 ··· βn 2 C with x = x1 xn , @ = @1 @n and aα,β , where n α = (α1; : : : ; αn); β = (β1; : : : ; βn) 2 N are multi-indices with N = f0; 1; 2;::: g and @i = @=@xi (i = 1;:::; n) denote derivations. Then what is a holonomic ideal? For a left ideal I of Dn, one can define the dimension of the left Dn-module Dn=I . One way is to define it as the (usual) dimension of the characteristic variety, which is an algebraic 2n subset of C . It is known that n ≤ dim Dn=I ≤ 2n if I =6 Dn. I is holonomic by definition if dim Dn=I = n or else I = Dn. Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Holonomic functions A function u = u(x) in the variables x = (x1;:::; xn) is called a holonomic function if its annihilator f 2 j g AnnDn u := P Dn Pu = 0 is a holonomic (left) ideal of Dn. For a left ideal I of Dn, one can define the dimension of the left Dn-module Dn=I . One way is to define it as the (usual) dimension of the characteristic variety, which is an algebraic 2n subset of C . It is known that n ≤ dim Dn=I ≤ 2n if I =6 Dn. I is holonomic by definition if dim Dn=I = n or else I = Dn. Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Holonomic functions A function u = u(x) in the variables x = (x1;:::; xn) is called a holonomic function if its annihilator f 2 j g AnnDn u := P Dn Pu = 0 is a holonomic (left) ideal of Dn. Then what is a holonomic ideal? . Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Holonomic functions A function u = u(x) in the variables x = (x1;:::; xn) is called a holonomic function if its annihilator f 2 j g AnnDn u := P Dn Pu = 0 is a holonomic (left) ideal of Dn. Then what is a holonomic ideal? For a left ideal I of Dn, one can define the dimension of the left Dn-module Dn=I . One way is to define it as the (usual) dimension of the characteristic variety, which is an algebraic 2n subset of C . It is known that n ≤ dim Dn=I ≤ 2n if I =6 Dn. I is holonomic by definition if dim Dn=I = n or else I = Dn. Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Integrals of holonomic functions If u(x1;:::; xn) is holonomic, then the integral Z v(x1;:::; xn−d ) := u(x1;:::; xn) dxn−d+1 ··· dxn; C where C is a d-cycle in Cd , or C = Rd , or C is a domain of Rd defined by polynomial inequalities, is also holonomic if the integral is `well-defined’. Moreover, a holonomic ideal ⊂ I AnnDn−d v is computable by using the D-module theoretic integration algorithm (O-Takayama (1999) for C without boundary, and O (to appear in JSC) for C with boundary defined by polynomials). (2) exp (g) with g 2 C(x) (by `localization algorithm' of O-Takayama-Walther 2000, or, more generally, `Weyl closure algorithm' by Tsai 2002). λ m (3) f (g0 + g1 log f + ··· + gm(log f ) ) with f ; g0;:::; gm 2 C[x], λ 2 C, m 2 N. (this talk). λ m I will show how to compute AnnDn f (log f ) . Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Examples of holonomic functions AnnDn u is precisely computable for the following u: λ1 ··· λm 2 C C (1) f1 fm with f1;:::; fm [x] = [x1;:::; xn] and λ1; : : : ; λm; especially a rational function (O 1997 for m = 1, O-Takayama 1999, Brian¸con-Maisonobe 2002 for m ≥ 1). λ m (3) f (g0 + g1 log f + ··· + gm(log f ) ) with f ; g0;:::; gm 2 C[x], λ 2 C, m 2 N. (this talk). λ m I will show how to compute AnnDn f (log f ) . Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Examples of holonomic functions AnnDn u is precisely computable for the following u: λ1 ··· λm 2 C C (1) f1 fm with f1;:::; fm [x] = [x1;:::; xn] and λ1; : : : ; λm; especially a rational function (O 1997 for m = 1, O-Takayama 1999, Brian¸con-Maisonobe 2002 for m ≥ 1). (2) exp (g) with g 2 C(x) (by `localization algorithm' of O-Takayama-Walther 2000, or, more generally, `Weyl closure algorithm' by Tsai 2002). Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Examples of holonomic functions AnnDn u is precisely computable for the following u: λ1 ··· λm 2 C C (1) f1 fm with f1;:::; fm [x] = [x1;:::; xn] and λ1; : : : ; λm; especially a rational function (O 1997 for m = 1, O-Takayama 1999, Brian¸con-Maisonobe 2002 for m ≥ 1). (2) exp (g) with g 2 C(x) (by `localization algorithm' of O-Takayama-Walther 2000, or, more generally, `Weyl closure algorithm' by Tsai 2002). λ m (3) f (g0 + g1 log f + ··· + gm(log f ) ) with f ; g0;:::; gm 2 C[x], λ 2 C, m 2 N. (this talk). λ m I will show how to compute AnnDn f (log f ) . Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix Examples of holonomic functions AnnDn u is precisely computable for the following u: λ1 ··· λm 2 C C (1) f1 fm with f1;:::; fm [x] = [x1;:::; xn] and λ1; : : : ; λm; especially a rational function (O 1997 for m = 1, O-Takayama 1999, Brian¸con-Maisonobe 2002 for m ≥ 1). (2) exp (g) with g 2 C(x) (by `localization algorithm' of O-Takayama-Walther 2000, or, more generally, `Weyl closure algorithm' by Tsai 2002). λ m (3) f (g0 + g1 log f + ··· + gm(log f ) ) with f ; g0;:::; gm 2 C[x], λ 2 C, m 2 N. (this talk). λ m I will show how to compute AnnDn f (log f ) . 3 − 2 s Example: Set f = x y . Then AnnDn[s]f is generated by 2 2y@x + 3x @y ; 2x@x + 3y@y − 6s: Remark: f (x)s is not a holonomic function in (x; s); for a fixed λ, f (x)λ is holonomic in x. Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix λ m Computing AnnDnf (log f ) Let f 2 K[x] = K[x1;:::; xn] be a non-constant polynomial with a computable subfield K of C. Then the annihilator of f s (log f )m with a parameter s can be computed as follows: s Step 1: Compute AnnDn[s]f . For f 2 K[x], consider the function f s = f (x)s with a s parameter s. Then AnnDn[s]f can be computed by an algorithm of O (1997) or of Brian¸on-Masionobe (2002). Remark: f (x)s is not a holonomic function in (x; s); for a fixed λ, f (x)λ is holonomic in x. Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix λ m Computing AnnDnf (log f ) Let f 2 K[x] = K[x1;:::; xn] be a non-constant polynomial with a computable subfield K of C. Then the annihilator of f s (log f )m with a parameter s can be computed as follows: s Step 1: Compute AnnDn[s]f . For f 2 K[x], consider the function f s = f (x)s with a s parameter s. Then AnnDn[s]f can be computed by an algorithm of O (1997) or of Brian¸on-Masionobe (2002). 3 − 2 s Example: Set f = x y . Then AnnDn[s]f is generated by 2 2y@x + 3x @y ; 2x@x + 3y@y − 6s: . Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix λ m Computing AnnDnf (log f ) Let f 2 K[x] = K[x1;:::; xn] be a non-constant polynomial with a computable subfield K of C. Then the annihilator of f s (log f )m with a parameter s can be computed as follows: s Step 1: Compute AnnDn[s]f . For f 2 K[x], consider the function f s = f (x)s with a s parameter s. Then AnnDn[s]f can be computed by an algorithm of O (1997) or of Brian¸on-Masionobe (2002). 3 − 2 s Example: Set f = x y . Then AnnDn[s]f is generated by 2 2y@x + 3x @y ; 2x@x + 3y@y − 6s: Remark: f (x)s is not a holonomic function in (x; s); for a fixed λ, f (x)λ is holonomic in x. Holonomic functions Annihilator with a parameter Specialization of the parameter s Application to integration Appendix s s s m Computing AnnDn[s](f ; f log f ;:::; f (log f ) ) Step 2: Differentiation with respect to s s Let G1 be a set of generators of AnnDn[s]f and e0 = (1; 0;:::; 0);:::; em = (0;:::; 0; 1) m+1 be the canonical basis of C .