New developments in Singular

Transcription

1 New developments in Singular H. Schönemann University of Kaiserslautern, Germany 2016/03/31 1 / 43

2 What is Singular? A computer algebra system for polynomial computations, with special emphasis on algebraic geometry, commutative and non-commutative algebra, singularity theory, and with packages for convex and tropical geometry. It is free and open-source under the GNU General Public Licence. 2 / 43

3 Open development model 3 / 43

4 Open development model Interaction with user base, bug & feature tracking Singular issue tracker 4 / 43

5 What is Singular? Over 30 development teams worldwide, over 170 libraries for advanced topics. 5 / 43

6 What is Singular? Singular consists of a kernel, written in C/C + +, and containing the core algorithms, libraries, written in the Singular language which provides a convenient way of user interaction and adding new mathematical features, and a comprehensive online manual and help function. 6 / 43

7 Singular Libraries 7 / 43

8 Example: Parametrizing Rational Curves Example > ring R = 0, (x,y,z), dp; > poly f = x5+10x4y+20x3y2+130x2y3-20xy4+20y5-2x4z-40x3yz-150x2y2z -90xy3z-40y4z+x3z2+30x2yz2+110xy2z2+20y3z2; > LIB "paraplanecurves.lib"; > genus(f); 0 > paraplanecurve(f); // paraplanecurve created a ring together with an ideal PARA. > def RP1 = paraplanecurve(f);setring RP1;PARA; PARA[1]=25s5-1025s4t+14825s3t s2t st t5 PARA[2]=25s5-1400s4t+30145s3t s2t st t5 PARA[3]=25s5-1025s4t+15575s3t s2t st t5 8 / 43

9 Example: Intersection Theory Example > LIB "schubert.lib"; > variety G = Grassmannian(2,4); > def r = G.baseRing; > setring r; > sheaf S = makesheaf(g,subbundle); > sheaf B = dualsheaf(s)^3; > integral(g,topchernclass(b)); 9 / 43

10 Example: Intersection Theory Example (continued) 27 Some keywords Schubert calculus, double point formulas, excess intersection formula, equivariant intersection theory using Bott s formula, Gromov-Witten invariants. 10 / 43

11 Key Algorithms in Singular Basic stuff Gröbner and standard Bases; free resolutions; polynomial factorization: Factory. More advanced stuff Primary decomposition: algorithms of Gianni-Trager-Zacharias, Shimoyama-Yokoyama, Eisenbud-Huneke-Vasconcelos: primdec.lib.; normalization: normal.lib, locnormal.lib, modnormal.lib. parametrization of rational plane curves: paraplanecurves.lib / 43

12 Parallelization: Processes processes, different memory areas Problem: data transfer can use a cluster of machines uses for Gröbner base computation: Chinese Remainder Theorem and Rational Reconstruction 12 / 43

13 Parallelization: Gröbner bases via Chinese Remainder Theorem choose (lucky) primes according to available CPU cores/machines compute a complete reduced Gröbner basis modulo these primes combine by rational reconstruction (Farey) 13 / 43

14 Parallelization: Threads threads - same process, common memory Problem: synchronization of data access uses for Gröbner base computation: Matrix operations (Gauss, F4) 14 / 43

15 Parallelization: Threads MathicGB (B. Roune): Gröbner basis computation via matrix operations matrix of machine size integers (modulo p) pivot parts used by all threads 15 / 43

16 Parallelization: Thread safe interpreter threads - same process, common memory Problem: synchronization of data access different memory domains 16 / 43

17 Example: Coarse Grained Parallelism in Singular Example > LIB"parallel.lib"; > LIB"random.lib"; > ring R = 0,x(1..4),dp; > ideal I = randomid(maxideal(3),3,100); > proc sizestd(ideal I, string monord) {def R = basering; list RL = ringlist(r); RL[3][1][1] = monord; def S = ring(rl); setring(s); return(size(std(imap(r,i))));} 17 / 43

18 Example: Coarse Grained Parallelism in Singular Example > list commands = " sizestd"," sizestd"; > list args = list(i,"lp"),list(i,"dp"); > parallelwaitfirst(commands, args); [1] empty list [2] 11 > parallelwaitall(commands, args); [1] 55 [2] / 43

19 Example: Adjoint Ideals We develop Parallel local-to-global algorithm computing G as intersection of local adjoint ideals. For C over Q, parallel modular approach with efficient verification test. Implementation in adjointideal.lib. Applications to computation of Riemann-Roch spaces (brillnoether.lib) rational parametrizations (paraplanecurves.lib). 19 / 43

20 Example: De Rham Cohomology General implementation of the algorithm of Oaku/Takayama/Walther for computing the de Rham cohomology of the complement U of a complex affine variety: derham.lib. De Rham cohomology of U is the hypercohomology of the de Rham complex on U, i.e. the complex of algebraic differential forms on U. Grothendieck and Deligne showed that it agrees with singular cohomology, hence, can be used to compute Betti numbers. Ongoing: Compute Gauss-Manin systems (direct images of D-modules), local monodromy. 20 / 43

21 New Libraries in Singular Framework for hyperplane arrangements: arr.lib; Riemann-Roch spaces: hess.lib, brillnoether.lib; Intersection theory: schubert.lib; New algorithms for computing tropical varieties: gfanlib.so; De Rham Cohomology: derham.lib; GIT-fans in geometric invariant theory: gitfan.lib. Parallel Computation of Gröbner Bases over Q (modstd.lib) and over algebraic function fields (ffmodstd.lib,nfmodstd.lib) Symbolic Computation with Chern Classes: chern.lib / 43

22 Connections to other Systems Singular Algebraic Geometry polymake Convex Geometry GAP Groups ANTIC Number Theory Included subsystems Factory: Polynomial Factorization; NTL: arithmetic for Number Theory; Flint: arithmetic for Number Theory; Plural: non-commutative stuff. 22 / 43

23 Homalg Example: From groups to vector bundles gap> LoadPackage( "repsn" );; gap> LoadPackage( "GradedModules" );; gap> G := SmallGroup( 1000, 93 ); <pc group of size 1000 with 6 generators> gap> Display( StructureDescription( G ) ); ((C5 x C5) : C5) : Q8 23 / 43

24 Homalg Example: From groups to vector bundles gap> V := Irr( G )[6];; Degree( V ); 5 gap> T0 := Irr( G )[5];; Degree( T0 ); 2 gap> T1 := Irr( G )[8];; Degree( T1 ); 5 gap> mu0 := ConstructTateMap( V, T0, T1, 2 ); <A homomorphism of graded left modules> 24 / 43

25 Homalg Example: From groups to vector bundles gap> A := HomalgRing( mu0 ); Q{e0,e1,e2,e3,e4} (weights: [ -1, -1, -1, -1, -1 ]) gap> M:=GuessModuleOfGlobalSectionsFromATateMap(2, mu0);; gap> ByASmallerPresentation( M ); <A graded non-zero module presented by 92 relations for 19 generators> 25 / 43

26 Homalg Example: From groups to vector bundles gap> S := HomalgRing( M ); Q[x0,x1,x2,x3,x4] (weights: [ 1, 1, 1, 1, 1 ]) gap> ChernPolynomial( M ); ( 2 1-h+4*h^2 ) -> P^4 gap> tate := TateResolution( M, -5, 5 );; 26 / 43

27 Homalg Example: From groups to vector bundles gap> Display( BettiTable( tate ) ); total: ???? : : * : * * : * * * : * * * * S twist: Euler: / 43

28 Computational Primary Decomposition: History algorithm by Wu in Maple (Wu-Ritt-process, 1989): via characteristic sets algorithm by Gianni,Traeger,Zacharias in AXIOM (1988) via factorization algorithm by Eisenbud, Huneke, Vasconcelos (1992) via equidimensional decomposition algorithm by Shimoyama, Yokoyama (1996) via characteristic series Singular implements GTZ, SY (1998), EHV (2001) 28 / 43

29 Computational Primary Decomposition by the Algorithm of Eisenbud, Huneke, Vasconcelos The algorithm by Eisenbud, Huneke, Vasconcelos decomposes into equidimensional parts Its main splitting tools is Proposition. If I R = K[x 1,..x n ] is an ideal, then the equidimensional hull of I E(I ) = AnnExt n d R (R/I, R), where d = dim(i ). 29 / 43

30 Computational Primary Decomposition by Characteristic Series Let < be the lexicographical ordering on R = K[x 1,..., x n ] with x 1 <... < x n. For f R let lvar(f ) (the leading variable of f ) be the largest variable in f, i.e., if f = a s (x 1,..., x k 1 )x s k a 0(x 1,..., x k 1 ) for some k n then lvar(f ) = x k. 30 / 43

31 Computational Primary Decomposition by Characteristic Series Let ini(f ) := a s (x 1,..., x k 1 ). The pseudoremainder r = prem(g, f ) of g with respect to f is defined by the equality ini(f ) a g = qf + r with deg lvar(f ) (r) < deg lvar(f ) (f ) and a minimal. A set T = {f 1,..., f r } R is called triangular if lvar(f 1 ) <... < lvar(f r ). Moreover, let U T, then (T, U) is called a triangular system, if T is a triangular set such that ini(t ) does not vanish on V (T ) \ V (U)(=: V (T \ U)). 31 / 43

32 Irreducible characteristic series T is called irreducible if for every i there are no d i,f i,f i such that lvar(d i ) < lvar(f i ) = lvar(f i ) = lvar(f i ), 0 prem({d i, ini(f i ), ini(f i )}, {f 1,..., f i 1 }), prem(d i f i f i f i, {f 1,..., f i 1 }) = 0. (T, U) is called irreducible if T is irreducible. 32 / 43

33 Irreducible characteristic series The main result on triangular sets is the following: Let G = {g 1,..., g s } R, then there are irreducible triangular sets T 1,..., T l such that V (G) = l i=1 (V (T i \ I i )) where I i = {ini(f ) f T i }. Such a set {T 1,..., T l } is called an irreducible characteristic series of the ideal (G). 33 / 43

34 Preprocessing: The Factorizing Buchberger Algorithm The factorizing Buchberger algorithm is the combination of Buchberger algorithm with factorization: each new element for the Gröbnerbasis will be factorized, and, if reducible, used to split the computation into several branches corresponding to the factors. Applied to an ideal I = (p 1,..., p s ) it computes a list of Gröbner bases G 1,..., G r such that V (I ) = V (G 1 )... V (G r ). The V (G i ) need not be irreducible, so this algorithm is mainly used as a preprocessing step. 34 / 43

35 Parallelizing: The Factorizing Buchberger Algorithm input: S = {s 1,..., s r } partial Gröbner basis D set of non-zero constraints (may be empty) L set of pairs etc. compute the Spoly h from L try to factor h if h does not factor: update S, L, continue h = h 1...h r : new sub-problems: S i = S {h i }, D i = D {h 1...h i 1 }, update L i check for subproblems describing the empty set: discard (Si, D i, L i ) if (D i ) (S i ) 35 / 43

36 Parallelizing: The Factorizing Buchberger Algorithm number of sub-problems vary all sub-problems are independent of each other work stealing can be used as the scheduling strategy to organize several workers to process the sub-problems 36 / 43

37 Zero-dimensional Primary Decomposition The lexicographical GB of a zero-dimensional ideal I contains one polynomial f of only the last variable. Let f α 1 αr 1...fr = f the decomposition of f in irreducible factors. Then the minimal primary decomposition of I is given by I = r k=1 (I, f α k k ) 37 / 43

38 Primary Decomposition: Reduction to Dimension 0 Proposition Let I K[x] = R be a proper ideal, and let u x be a subset of maximal cardinality such that I K[u] = {0}. Then: The ideal I K(u)[x \u] K(u)[x \u] is zero-dimensional. Let > = (> x\u, > u ) be a global product ordering on K[x], and let G be a Gröbner basis for I with respect to >. Then G is a Gröbner basis for I K(u)[x \u] with respect to the monomial ordering obtained by restricting > to the monomials in K[x \u]. 38 / 43

39 Primary Decomposition: Reduction to Dimension 0 If h K[u] is the least common multiple of the leading coefficients of the elements of G (regarded as polynomials in K(u)[x \u]), then I K(u)[x \u] K[x] = I : h. 39 / 43

40 Primary Decomposition: Reduction to Dimension 0 Proposition Let I K[x] = R be a proper ideal, and let u x be a subset of maximal cardinality such that I K[u] = {0}. Then: The ideal I K(u)[x \u] K(u)[x \u] is zero-dimensional. Let > = (> x\u, > u ) be a global product ordering on K[x], and let G be a Gröbner basis for I with respect to >. Then G is a Gröbner basis for I K(u)[x \u] 40 / 43

41 Primary Decomposition: Reduction to Dimension 0 All primary components of the ideal I K(u)[x \u] K[x] have the same dimension, namely dim I. If I K(u)[x \u] = Q 1... Q r is the minimal primary decomposition, then I K(u)[x \u] K[x] = (Q 1 K[x])... (Q r K[x]) is the minimal primary decomposition, too. By recursion, the proposition allows us to reduce the general case of primary decomposition to the zero-dimensional case. In turn, if I K[x] is a zero-dimensional ideal in general position (with respect to the lexicographic order satisfying x 1 > > x n ), and if h n is a generator for I K[x n ], the minimal primary decomposition of I is obtained by factorizing h n. In characteristic zero, the condition that I is in general position can be achieved by means of a generic linear coordinate transformation 41 / 43

42 Primary decomposition: Summary of basic operations Gröbner basis computation factorizing polynomials over K resp K(x)) ideal operations: intersection, quotient, saturation coordinate transformation 42 / 43

43 Thank you 43 / 43