New developments in Singular

H. SchönemannUniversity of Kaiserslautern, Germany

2016/03/31

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What is Singular?A computer algebra system for polynomial computations, with specialemphasis 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.

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Open development model

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Open development model

Singular issue tracker

Interaction with user base, bug & feature tracking

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What is Singular?

Over 30 development teams worldwide, over 170 libraries for advancedtopics.

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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 aconvenient way of user interaction and adding new mathematicalfeatures, and

a comprehensive online manual and help function.

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Singular Libraries

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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+14825s3t2-85565s2t3+146420st4-20780t5

PARA[2]=25s5-1400s4t+30145s3t2-305650s2t3+1396410st4-2023972t5

PARA[3]=25s5-1025s4t+15575s3t2-116205s2t3+562440st4-1898652t5

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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));

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Example: Intersection Theory

Example (continued)

27

Some keywords

Schubert calculus, double point formulas, excess intersection formula,equivariant intersection theory using Bott’s formula, Gromov-Witteninvariants.

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Key Algorithms in SingularBasic 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

...

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Parallelization: Processes

processes, different memory areasProblem: data transfer

can use a cluster of machines

uses for Gröbner base computation:Chinese Remainder Theorem and Rational Reconstruction

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Parallelization: Gröbner bases via Chinese RemainderTheorem

choose (lucky) primes according to available CPU cores/machines

compute a complete reduced Gröbner basis modulo these primes

combine by rational reconstruction (Farey)

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Parallelization: Threads

threads - same process, common memoryProblem: synchronization of data access

uses for Gröbner base computation:Matrix operations (Gauss, F4)

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Parallelization: Threads

MathicGB (B. Roune): Gröbner basis computation via matrixoperations

matrix of machine size integers (modulo p)

pivot parts used by all threads

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Parallelization: Thread safe interpreter

threads - same process, common memoryProblem: synchronization of data access

different memory domains

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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))));}

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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] 11

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Example: Adjoint Ideals

We develop

Parallel local-to-global algorithm computing G as intersection of localadjoint ideals.

For C over Q, parallel modular approach with efficient verificationtest.

Implementation in adjointideal.lib.

Applications to computation of

Riemann-Roch spaces (brillnoether.lib)

rational parametrizations (paraplanecurves.lib).

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Example: De Rham Cohomology

General implementation of the algorithm of Oaku/Takayama/Waltherfor computing the de Rham cohomology of the complement U of acomplex affine variety: deRham.lib.

De Rham cohomology of U is the hypercohomology of the de Rhamcomplex on U, i.e. the complex of algebraic differential forms on U.

Grothendieck and Deligne showed that it agrees with singularcohomology, hence, can be used to compute Betti numbers.

Ongoing: Compute Gauss-Manin systems (direct images ofD-modules), local monodromy.

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New Libraries in SingularFramework 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) andover algebraic function fields (ffmodstd.lib,nfmodstd.lib)

Symbolic Computation with Chern Classes: chern.lib

...

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Connections to other Systems

GAPGroups

SingularAlgebraic Geometry

polymakeConvex Geometry

ANTICNumber Theory

Included subsystems

Factory: Polynomial Factorization;

NTL: arithmetic for Number Theory;

Flint: arithmetic for Number Theory;

Plural: non-commutative stuff.

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Homalg Example: From groups to vector bundles

gap> LoadPackage( "repsn" );;

gap> LoadPackage( "GradedModules" );;

gap> G := SmallGroup( 1000, 93 );

gap> Display( StructureDescription( G ) );

((C5 x C5) : C5) : Q8

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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 );

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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 );

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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 );;

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Homalg Example: From groups to vector bundles

gap> Display( BettiTable( tate ) );

total: 100 37 14 10 5 2 5 10 14 37 100 ? ? ? ?

----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

4: 100 35 4 . . . . . . . . 0 0 0 0

3: * . 2 10 10 5 . . . . . . 0 0 0

2: * * . . . . . 2 . . . . . 0 0

1: * * * . . . . . . 5 10 10 2 . 0

0: * * * * . . . . . . . . 4 35 100

----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---S

twist: -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5

-------------------------------------------------------------------

Euler: 100 35 2 -10 -10 -5 0 2 0 -5 -10 -10 2 35 100

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Computational Primary Decomposition: History

algorithm by Wu in Maple (Wu-Ritt-process, 1989): via characteristicsets

algorithm by Gianni,Traeger,Zacharias in AXIOM (1988) viafactorization

algorithm by Eisenbud, Huneke, Vasconcelos (1992) viaequidimensional decomposition

algorithm by Shimoyama, Yokoyama (1996) via characteristic series

Singular implements GTZ, SY (1998), EHV (2001)

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Computational Primary Decomposition by the Algorithm ofEisenbud, Huneke, Vasconcelos

The algorithm by Eisenbud, Huneke, Vasconcelos decomposes intoequidimensional parts Its main splitting tools isProposition. If I ⊆ R = K [x1, ..xn] is an ideal, then the equidimensionalhull of I E (I ) = AnnExtn−dR (R/I ,R), where d = dim(I ).

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Computational Primary Decomposition by CharacteristicSeries

Let < be the lexicographical ordering on R = K [x1, ..., xn] withx1 < ... < xn. For f ∈ R let lvar(f ) (the leading variable of f ) be thelargest variable in f , i.e., if f = as(x1, ..., xk−1)x

sk + ... + a0(x1, ..., xk−1) for

some k ≤ n then lvar(f ) = xk .

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Computational Primary Decomposition by CharacteristicSeries

Let ini(f ) := as(x1, ..., xk−1). The pseudoremainder r = prem(g , f ) of gwith respect to f is defined by the equality ini(f )a · g = qf + r withdeglvar(f )(r) < deglvar(f )(f ) and a minimal. A set T = {f1, ..., fr} ⊂ R iscalled triangular if lvar(f1) < ... < lvar(fr ). Moreover, let U ⊂ T , then(T ,U) is called a triangular system, if T is a triangular set such thatini(T ) does not vanish on V (T ) \ V (U)(=: V (T \ U)).

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Irreducible characteristic series

T is called irreducible if for every i there are no di ,f′i ,f′′i such that

lvar(di ) < lvar(fi ) = lvar(f′i ) = lvar(f

′′i ),

0 6∈ prem({di , ini(f ′i ), ini(f ′′i )}, {f1, ..., fi−1}),

prem(di fi − f ′i f ′′i , {f1, ..., fi−1}) = 0.

(T ,U) is called irreducible if T is irreducible.

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Irreducible characteristic series

The main result on triangular sets is the following: LetG = {g1, ..., gs} ⊂ R, then there are irreducible triangular sets T1, ...,Tlsuch that V (G ) =

⋃li=1(V (Ti \ Ii )) where Ii = {ini(f ) | f ∈ Ti}. Such a

set {T1, ...,Tl} is called an irreducible characteristic series of the ideal(G ).

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Preprocessing: The Factorizing Buchberger Algorithm

The factorizing Buchberger algorithm is the combination of Buchbergeralgorithm with factorization: each new element for the Gröbnerbasis willbe factorized, and, if reducible, used to split the computation into severalbranches corresponding to the factors. Applied to an ideal I = (p1, ..., ps)it computes a list of Gröbner bases G1, ...,Gr such that

V (I ) = V (G1) ∪ ... ∪ V (Gr )

.The V (Gi ) need not be irreducible, so this algorithm is mainly used as apreprocessing step.

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Parallelizing: The Factorizing Buchberger Algorithm

input: S = {s1, ..., sr} partial Gröbner basisD set of non-zero constraints (may be empty)L set of pairs etc.

compute the Spoly h from L

try to factor hI if h does not factor: update S , L, continueI h = h1...hr : new sub-problems: Si = S ∪ {hi}, Di = D ∪ {h1...hi−1},

update LiI check for subproblems describing the empty set: discard (Si ,Di , Li ) if

(Di ) ⊆ (Si )

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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 organizeseveral workers to process the sub-problems

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Zero-dimensional Primary Decomposition

The lexicographical GB of a zero-dimensional ideal I contains onepolynomial f of only the last variable. Let f α11 ....f

αrr = f the

decomposition of f in irreducible factors.Then the minimal primary decomposition of I is given by

I = ∩rk=1(I , fαkk )

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Primary Decomposition: Reduction to Dimension 0

Proposition Let I ( K [x ] = R be a proper ideal, and let u ⊂ x be asubset 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 Gbe a Gröbner basis for I with respect to >. Then G is a Gröbner basisfor I K (u)[x \u ] with respect to the monomial ordering obtained byrestricting > to the monomials in K [x \u ].

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Primary Decomposition: Reduction to Dimension 0

If h ∈ K [u] is the least common multiple of the leading coefficients of theelements of G (regarded as polynomials in K (u)[x \u ]), then

I K (u)[x \u ] ∩ K [x ] = I : 〈h〉∞ .

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Primary Decomposition: Reduction to Dimension 0

Proposition Let I ( K [x ] = R be a proper ideal, and let u ⊂ x be asubset 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 Gbe a Gröbner basis for I with respect to >. Then G is a Gröbner basisfor I K (u)[x \u ]

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Primary Decomposition: Reduction to Dimension 0

All primary components of the ideal I K (u)[x \u ] ∩ K [x ] have thesame dimension, namely dim I . If I K (u)[x \u ] = Q1 ∩ . . . ∩ Qr is theminimal primary decomposition, then

I K (u)[x \u ] ∩ K [x ] = (Q1 ∩ K [x ]) ∩ . . . ∩ (Qr ∩ K [x ])

is the minimal primary decomposition, too.

By recursion, the proposition allows us to reduce the general case ofprimary decomposition to the zero-dimensional case. In turn, if I ⊂ K [x ] isa zero-dimensional ideal “in general position” (with respect to thelexicographic order satisfying x1 > · · · > xn), and if hn is a generator forI ∩ K [xn], the minimal primary decomposition of I is obtained byfactorizing hn. In characteristic zero, the condition that I is in generalposition can be achieved by means of a generic linear coordinatetransformation

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Primary decomposition: Summary of basic operations

Gröbner basis computation

factorizing polynomials over K resp K (x))

ideal operations: intersection, quotient, saturation

coordinate transformation

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Thank you

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