An Improvement of Rosenfeld-Gröbner Algorithm

Transcription

1 An Improvement of Rosenfeld-Gröbner Algorithm Amir Hashemi 1,2 Zahra Touraji 1 1 Department of Mathematical Sciences Isfahan University of Technology Isfahan, Iran 2 School of Mathematics Institute for Research in Fundamental Sciences (IPM) Tehran, Iran ICMS 14, August 5-9, Hanyang University, Seoul, Korea A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

2 Hakim Abolfath Omar ebn Ibrahim Khayyám Nieshapuri (18 May December 1131), born in Nishapur in North Eastern Iran, was a great Persian polymath, philosopher, mathematician, astronomer and poet, who wrote treatises on mechanics, geography, mineralogy, music, and Islamic theology Khayyám was famous during his times as a mathematician He wrote the influential treatise on demonstration of problems of algebra (1070), which laid down the principles of algebra, part of the body of Persian Mathematics that was eventually transmitted to Europe In particular, he derived general methods for solving cubic equations and even some higher orders A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

3 He wrote on the triangular array of binomial coefficients known as Khayyám-Pascal s triangle In 1077, Khayyám also wrote a book published in English as On the Difficulties of Euclid s Definitions An important part of the book is concerned with Euclid s famous parallel postulate and his approach made their way to Europe, and may have contributed to the eventual development of non-euclidean geometry A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

4 Outline of talk 1 Introduction Differential algebra 2 Buchberger s first criterion 3 Rosenfeld-Gröbner algorithm 4 Example A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

5 Differential algebra Differential algebra Algebra techniques to analyze PDE s A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

6 Differential algebra Differential algebra Algebra techniques to analyze PDE s It was initiated mostly by French and American researchers and developed by the American teams J F Ritt ( ) -which is the father of differential algebra- and E R Kolchin ( ); a PhD student of Ritt The aim is to study systems of polynomial nonlinear ordinary and partial differential equations (PDE s) from a purely algebraic point of view Differential algebra is closer to ordinary commutative algebra than to analysis A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

7 Notations Introduction Buchberger s first criterion K; an arbitrary field R = K[x 1,, x n ]; polynomial ring in the x i s F = {f 1,, f k } R; a set of polynomials I = F = { k i=1 p if i p i R}; the ideal generated by the f i s A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

8 Some important monomial orderings Buchberger s first criterion Lex(Lexicographic) Ordering X α lex X β if leftmost nonzero of α β is < 0 Degree Reverse Lex Ordering X α drl X β if α < β or α = β and rightmost nonzero of α β is > 0 Example x 100 lex(y,x) y y drl(y,x) x 100 A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

9 Further notations Introduction Buchberger s first criterion R = K[x 1,, x n ], f R a monomial ordering on R I R an ideal LM(f): The greatest monomial (with respect to ) in f 5x 3 y 2 + 4x 2 y 3 + xy + 1 LC(f): The coefficient of LM(f) in f 5x 3 y 2 + 4x 2 y 3 + xy + 1 LT(f): LC(f)LM(f) 5x 3 y 2 +4x 2 y 3 + xy + 1 LT(I): LT(f) f I A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

10 Buchberger s first criterion Definition I K[x 1,, x n ] A monomial ordering A finite set G I is a Gröbner Basis for I wrt, if LT(I) = LT(g) g G Existence of Each ideal has a Gröbner basis Example I = xy x, x 2 y, x drl y LT(I) = x 2, xy, y 2 A Gröbner basis is: {xy x, x 2 y, y 2 y} A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

11 Buchberger s first criterion Buchberger s Criterion and Buchberger s Algorithm Definition S-polynomial Spoly(f, g) = xγ LT(f) f xγ LT(g) g x γ = lcm(lm(f), LM(g)) Spoly(x 3 y 2 +xy 3, xyz z 3 ) = z(x 3 y 2 +xy 3 ) x 2 y(xyz z 3 ) = zxy 3 +x 2 yz 3 Buchberger s Criterion G is a Gröbner basis for G g i, g j G, remainder((spoly(g i, g j ), G) = 0 A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

12 Buchberger s first criterion Theorem Suppose that f, g K[x 1,, x n ]are two polynomials so that lcm(lm(f), LM(g)) = LM(f)LM(g) Then, the remainder of the division of Spoly(f, g) by {f, g} is 0 Buchberger s Criterion G is a Gröbner basis for G g i, g j G which do not satisfy above criterion we have remainder((spoly(g i, g j ), G) = 0 Example Let f = x 2 + x and g = y 2 + y Then, {f, g} is a Gröbner basis of f, g wrt any ordering A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

13 Rosenfeld-Gröbner algorithm Definition An operator δ : R R is a derivation operator, if δ(a + b) = δ(a) + δ(b) δ(ab) = δ(a)b + aδ(b) A differential ring is a pair (R, ) satisfying = {δ 1,, δ m } δ i δ j a = δ j δ i a If m = 1, then R is called an ordinary differential ring; otherwise it will be called partially Definition An algebraic ideal I of R is called a differential ideal if δa I for each δ and a I A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

14 Example Introduction Rosenfeld-Gröbner algorithm C[x 1,, x m ] together with the set of operators / x 1,, / x m is a differential ring A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

15 Rosenfeld-Gröbner algorithm Example C[x 1,, x m ] together with the set of operators / x 1,, / x m is a differential ring Definition Θ := { δ t 1 1 δ t 2 2 δ tm m t 1,, t m N } θ = δ t 1 1 δ tm m Θ is called a derivation operator of R ord(θ) := m i=1 t i is called the order of θ a differential polynomial ring: R := K{u 1,, u n } := K [ ΘU ] is the usual commutative polynomial ring generated by ΘU over K, where U := {u 1,, u n } is the set of differential indeterminate A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

16 Ranking Introduction Rosenfeld-Gröbner algorithm R = K{u 1,, u n } : a differential polynomial ring U = {u 1,, u n } Θ := { δ t 1 1 δ t 2 2 δm tm t 1,, t m N } Θ := { δ t 1 1 δ t 2 2 δ t m u t 1,, t m N, u U } Definition ranking >: δ Θ, v, w ΘU : δv > v and v > w = δv > δw Example R := Q(x, y){u, v}, v < lex u, y < lex x p := vx,y,yu 2 x,y + v x u 2 x,x + u x ; leader(p)=u x,x ; rank(p)=u 2 x,x initial(p)=v x separant(p):= p/ leader(p) = 2u x,x A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

17 Rosenfeld-Gröbner algorithm Definition Let us consider two differential polynomials p 1 and p 2 with ld(p i ) = θ i u i, i = 1, 2 (p 1, p 2 ) = { s p2 θ 1,2 θ 1 p 1 s p1 θ 1,2 θ 2 p 2 u 1 = u 2, 0 u 1 u 2, where θ 1,2 = lcm(θ 1, θ 2 ) Example R := Q(x, y){u, v}, v < lex u, y < lex x p := u y + v; q := vu 2 x + v y,y ; (p, q) = 2vu x v x v y ux 2 v y,y,y A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

18 Rosenfeld-Gröbner algorithm Rosenfeld-Gröbner algorithm: Constructing the -polynomials Reducing each -polynomial (partially and algebraically) wrt the current set of polynomials Discussing the nullity of initials ( 2 u x ) 2 v + 2 u 2 x v + u 2 x = 0 Σ : 2 u x y = 0 2 u y 2 1 = 0 A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

19 Rosenfeld-Gröbner algorithm Rosenfeld-Gröbner algorithm: Constructing the -polynomials Reducing each -polynomial (partially and algebraically) wrt the current set of polynomials Discussing the nullity of initials ( 2 u x ) 2 v + 2 u 2 x v + u 2 x = 0 Σ : 2 u x y = 0 2 u y 1 = 0 Step 1: Defining a 2 differential ring: Q(x, y)[u, v] A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

20 Rosenfeld-Gröbner algorithm Rosenfeld-Gröbner algorithm: Constructing the -polynomials Reducing each -polynomial (partially and algebraically) wrt the current set of polynomials Discussing the nullity of initials ( 2 u x ) 2 v + 2 u 2 x v + u 2 x = 0 Σ : 2 u x y = 0 2 u y 1 = 0 Step 1: Defining a 2 differential ring: Q(x, y)[u, v] Step 2: Constructing the differential ideal I = u 2 x,xv + u x,x v + u x, u x,y, u 2 y,y 1 A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

21 Rosenfeld-Gröbner algorithm Rosenfeld-Gröbner algorithm: Constructing the -polynomials Reducing each -polynomial (partially and algebraically) wrt the current set of polynomials Discussing the nullity of initials ( 2 u x ) 2 v + 2 u 2 x v + u 2 x = 0 Σ : 2 u x y = 0 2 u y 1 = 0 Step 1: Defining a 2 differential ring: Q(x, y)[u, v] Step 2: Constructing the differential ideal I = u 2 x,xv + u x,x v + u x, u x,y, u 2 y,y 1 Step 3: Decomposing I by Rosenfeld-Gröbner Algorithm: I = u 2 y,y 1, u x u 2 y,y 1, 4u x v, v x + 2, v y u 2 x,xv +u x,x v +u x, u x,y, u 2 y,y 1, v y : v, 2u x,x +1 A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

22 Rosenfeld-Gröbner algorithm u 2 x,xv + u x,x v + u x, u x,y, u 2 y,y 1, v y : v, 2u x,x + 1 We impose the following initial values to find the Taylor series up to order 3: v(x, y) = 1 + xf(x) and, u(0, 0) = c 0, 2 u y 2 (0, 0) = 1, 2 u u (0, 0) = 1, x2 y (0, 0) = c 1, u (0, 0) = 0 x A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

23 Rosenfeld-Gröbner algorithm u 2 x,xv + u x,x v + u x, u x,y, u 2 y,y 1, v y : v, 2u x,x + 1 We impose the following initial values to find the Taylor series up to order 3: v(x, y) = 1 + xf(x) and, u(0, 0) = c 0, 2 u y 2 (0, 0) = 1, 2 u u (0, 0) = 1, x2 y (0, 0) = c 1, u (0, 0) = 0 x { u(x, y) = c0 + c 1 y + 1/2y 2 1/2x 2 1/6x 3 v(x, y) = 1 + f(0)x + f (0)x 2 + 1/2f (0)x 3 A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

24 Rosenfeld-Gröbner algorithm Carra-Ferro and Ollivier, 87 : Developing the concept of differential Gröbner bases Mansfield, 91 : Developing another concept of differential Maple package diffgrob2 to solve systems of linear or nonlinear PDEs Boulier et al, 09 : Presenting Rosenfeld-Gröbner algorithm Maple package DifferentialAlgebra A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

25 Example Theorem (Boulier et al, 09) Let p and q be two differential polynomials which are linear, homogeneous, in one differential indeterminate and with constant coefficients Further, suppose that we have lcm(θu, ϕu) = θϕu where ld(p) = θu and ld(q) = ϕu Then the full differential remainder of (p, q) wrt {p, q} is zero Example R := Q(x, y){u}, y < lex x p := u x,x + u x,y ; x 2 + xy q := u y,y + u y, y 2 + y; A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

26 Example Theorem Let p and q be two differential polynomials which are products of differential polynomials which are linear, homogeneous, in the same differential indeterminate and with constant coefficients Furthermore, suppose that we have lcm(θu, ϕu) = θϕu where ld(p) = θu and ld(q) = ϕu Then the full differential remainder of (p, q) wrt {p, q} is zero Maple package RosenfeldGrobnermpl: Example R := Q(x, y){u}, y < lex x p := (u x,x,x,x u) 2 (u x,x + u x ) 3 q := (u y,y,y u y,y ) 2 (u y + u) 3 Time and memory by DifferentialAlgebra: 258 sec, 88Gb Time and memory by RosenfeldGrobnermpl: 172 sec, 84Gb A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm

27 Example Thanks for your attention! A Hashemi & Z Touraji An Improvement of Rosenfeld-Gröbner Algorithm