SINGULAR and Applications
talk at the
CIMPA School Lahore 2012
Gerhard Pfister
[email protected]
Departement of Mathematics
University of Kaiserslautern
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 1
SINGULAR
A Computer Algebra System for Polynomial Computations
with special emphasize on the needs of algebraic geometry, commutative algebra, and
singularity theory
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SINGULAR
A Computer Algebra System for Polynomial Computations
with special emphasize on the needs of algebraic geometry, commutative algebra, and
singularity theory
W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann
Technische Universität Kaiserslautern
Fachbereich Mathematik; Zentrum für Computer Algebra
D-67663 Kaiserslautern
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SINGULAR
A Computer Algebra System for Polynomial Computations
with special emphasize on the needs of algebraic geometry, commutative algebra, and
singularity theory
W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann
Technische Universität Kaiserslautern
Fachbereich Mathematik; Zentrum für Computer Algebra
D-67663 Kaiserslautern
The computer is not the philosopher’s stone but the philosopher’s
whetstone
Hugo Battus, Rekenen op taal 1983
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Twisted cubic curve
f1 , . . . , fm ∈ C[x1 , . . . , xn ]
V (f1 , . . . , fm ) = {p ∈ Cn such that fi = 0 for all i}
the white curve V (y − x2 , xy − z) is the intersection of the red
surface V (xy − z) and the blue surface V (y − x2 )
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Cusp
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Cup
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A1
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Cayley-cubic
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Heart
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Adding machine 1623
Wilhelm Schickard (Universität Tübingen)
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Slide rule
Let us compute 1, 3 ∗ 4, 2 .
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Adding machine
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log table
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Logarithm
√
Let us compute 3 5 :
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 13
Logarithm
√
Let us compute 3 5 :
√ 1
3
5 = · log(5)
log
3
log(5) = 0, 6990
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 13
Logarithm
√
Let us compute 3 5 :
√ 1
3
5 = · log(5)
log
3
log(5) = 0, 6990
1
· log(5) = 0, 2330 = log(1, 71)
3
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 13
Logarithm
√
Let us compute 3 5 :
√ 1
3
5 = · log(5)
log
3
log(5) = 0, 6990
1
· log(5) = 0, 2330 = log(1, 71)
3
1, 713 = 5, 0002
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 13
Punch card
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ZXSpectrum
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K1630
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Atari
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Birth of SINGULAR
1984
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History
1983
Greuel/Pfister: Exist singularities (not quasi-homogeneous and
complete intersection) with exact Poincaré-complex?
1984
Neuendorf/Pfister: Implementation of the Gröbner basis algorithm in basic at ZX-Spectrum
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 19
History
1983
Greuel/Pfister: Exist singularities (not quasi-homogeneous and
complete intersection) with exact Poincaré-complex?
1984
Neuendorf/Pfister: Implementation of the Gröbner basis algorithm in basic at ZX-Spectrum
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 19
History
1983
Greuel/Pfister: Exist singularities (not quasi-homogeneous and
complete intersection) with exact Poincaré-complex?
1984
Neuendorf/Pfister: Implementation of the Gröbner basis algorithm in basic at ZX-Spectrum
2002
Book: A SINGULAR Introduction to Commutative Algebra
(G.-M. Greuel and G. Pfister, with contributions by O. Bachmann,
C. Lossen and H. Schönemann).
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 19
History
2004
Jenks Price
for:
Excellence in Software Engineering
awarded at ISSAC in Santander
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History
2004
Jenks Price
for:
Excellence in Software Engineering
awarded at ISSAC in Santander
http://www.singular.uni-kl.de
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History
2004
Jenks Price
for:
Excellence in Software Engineering
awarded at ISSAC in Santander
http://www.singular.uni-kl.de
Supported by: Deutsche Forschungsgemeinschaft, Stiftung Rheinland-Pfalz für
Innovation, Volkswagen Stiftung
SINGULAR is free software (Gnu Public Licence)
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Team
G. Pfister,H. Schönemann, W. Decker,G.-M. Greuel
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Team
Kaiserslautern
Saarbrücken
Cottbus
Berlin
Mainz
Dortmund
Valladolid
La Laguna
Buenos Aires
Cape Town
...
G. Pfister,H. Schönemann, W. Decker,G.-M. Greuel
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Teams
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Saito’s result
Theorem (K. Saito 1971): Let (X, 0) be the germ of an isolated
hypersurface singularity. The following conditions are equivalent:
(X, 0) is quasi-homogeneous.
µ(X, 0) = τ (X, 0).
The Poincaré complex of (X, 0) is exact.
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Saito’s result
Theorem (K. Saito 1971): Let (X, 0) be the germ of an isolated
hypersurface singularity. The following conditions are equivalent:
(X, 0) is quasi-homogeneous.
µ(X, 0) = τ (X, 0).
The Poincaré complex of (X, 0) is exact.
We wanted to generalize this theorem to the case of
isolated complete intersection singularities.
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Poincaré complex
Let (Xl,k , 0) be the germ of the unimodal space curve singularity
F Tk,l of the classification of Terry Wall defined by the equations
xy + z l−1 = 0
xz + yz 2 + y k−1 = 0
4 ≤ l ≤ k, 5 ≤ k .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 24
Poincaré complex
Let (Xl,k , 0) be the germ of the unimodal space curve singularity
F Tk,l of the classification of Terry Wall defined by the equations
xy + z l−1 = 0
xz + yz 2 + y k−1 = 0
The Poincaré complex
4 ≤ l ≤ k, 5 ≤ k .
0 −→ C −→ OXl,k ,0 −→ Ω1Xl,k ,0 −→ Ω2Xl,k ,0 −→ Ω3Xl,k ,0 −→ 0
is exact.
But (Xl,k , 0) is not quasi-homogeneous:
µ(X, 0) = τ (X, 0) + 1 = k + l + 2.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 24
µ and τ
Let (X, 0) be a germ of a space curve singularity defined by
f = g = 0, with f, g ∈ C{x, y, z}
µ(X, 0) = dimC (Ω1X,0 /dO(X,0) )
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 25
µ and τ
Let (X, 0) be a germ of a space curve singularity defined by
f = g = 0, with f, g ∈ C{x, y, z}
µ(X, 0) = dimC (Ω1X,0 /dO(X,0) )
τ (X, 0) = dimC (C{x, y, z}/ < f, g, M1 , M2 , M3 >)
here the Mi are the 2-minors of the Jacobian matrix of f, g .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 25
µ and τ
Let (X, 0) be a germ of a space curve singularity defined by
f = g = 0, with f, g ∈ C{x, y, z}
µ(X, 0) = dimC (Ω1X,0 /dO(X,0) )
τ (X, 0) = dimC (C{x, y, z}/ < f, g, M1 , M2 , M3 >)
here the Mi are the 2-minors of the Jacobian matrix of f, g .
Reiffen: The Poincaré complex is exact if and only if
< f, g > Ω3C3 ,0 ⊂ d(< f, g > Ω2C3 ,0 )
and
µ(X, 0) = dimC (Ω2X,0 ) − dimC (Ω3X,0 )
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 25
Poincaré complex
Let A = C[x1 , . . . , xn ] or the localization of the polynomial ring in
the ideal < x1 , . . . , xn >
Let M ⊂ Am be a sub-module. We need to compute the
C-vectorspace dimension of Am /M .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 26
Poincaré complex
Let A = C[x1 , . . . , xn ] or the localization of the polynomial ring in
the ideal < x1 , . . . , xn >
Let M ⊂ Am be a sub-module. We need to compute the
C-vectorspace dimension of Am /M .
dimC (C[x, y]/ < y 3 , x2 y, x4 >= 8,
because 1, x, y, x2 , xy, y 2 , x3 , xy 2 is a basis.
dimC (C[x, y]/ < y 3 + x2 , y 4 + xy 2 >= 8
is not so easy to see.
We need Gröbner bases to see this.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 26
Poincaré complex: SINGULAR session
ring R=0,(x,y),dp;
ideal I=y3+x2,y4+xy2;
ideal J=std(I);
vdim(J);
8
J;
J[1℄=y3+x2
J[2℄=x2y-xy2
J[3℄=x4+x3
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 27
Elimination
Polynomials and Power Series,
May They Forever Rule the World
Shreeram S. Abhyankar
Polynomials and power series.
May they forever rule the world.
Eliminate, eliminate, eliminate.
Eliminate the eliminators of elimination theory.
As you must resist the superbourbaki coup,
so must you fight the little bourbakis too.
Kronecker, Kronecker, Kronecker above all
Kronecker, Mertens, Macaulay, and Sylvester.
Not the theology of Hilbert,
But the constructions of Gordon.
Not the surface of Riemann,
But the algorithm of Jacobi.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 28
Elimination
Ah! the beauty of the identity of Rogers and Ramanujan!
Can it be surpassed by Dirichlet and his principle?
Germs, viruses, fungi, and functors,
Stacks and sheaves of the lot
Fear them not
We shall be victors.
Come ye forward who dare present a functor,
We shall eliminate you
By resultants, discriminants, circulants and alternants.
Given to us by Kronecker, Mertens, Sylvester.
Let not here enter the omologists, homologists,
And their cohorts the cohomologists crystalline
For this ground is sacred.
Onward Soldiers! defend your fortress,
Fight the Tor with a determinant long and tall,
But shun the Ext above all.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 29
Elimination
Morphic injectives, toxic projectives,
Etal, eclat, devious devisage,
Arrows poisonous large and small
May the armor of Tschirnhausen
Protect us from the scourge of them all.
You cannot conquer us with rings of Chow
And shrieks of Chern
For we, too, are armed with polygons of Newton
And algorithms of Perron.
To arms, to arms, fractions, continued or not,
Fear not the scheming ghost of Grothendieck
For the power of power series is with you,
May they converge or not
(May they be polynomials or not)
(May they terminate or not).
Can the followers of G by mere “smooth” talk
Ever make the singularity simple?
Long live Karl Weierstrass!
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 30
Elimination
What need have we for rings Japanese, excellent or bad,
When, in person, Nagata himself is on our side.
What need to tensorize
When you can uniformize,
What need to homologize
When you can desingularize
(Is Hironaka on our side?)
Alas! Princeton and fair Harvard you, too,
Reduced to satellite in the Bur-Paris zoo.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 31
Elimination
Marlis -Problem:
Find the equations of the curve given by the parametrization:
x = −t5 + t4 + t3 − 2t2 + 1
y = −t3 − t2 + 1
z = −t5 + t3
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 32
Elimination
ring r=0,(x,y,z,t),dp;
ideal I=
x-(1-2t2+t3+t4-t5),
y-(1-t2-t3),
z-(t3-t5);
ideal J=eliminate(I,t);
J;
J[1℄=x2z-xyz+2y2z-2xz2+z3+3x2-8xy+y2-11xz+5yz+12z2+5x-5z-1;
J[2℄=y3+x2-4xy-3xz+6yz+2z2+3x-y-6z;
J[3℄=xy2-y2z-x2+4xz-4z2;
J[4℄=x2y-2xyz-y2z+yz2-3x2+3xy+10xz-3yz-8z2-x+2z;
J[5℄=x3-4xyz+5y2z-3xz2+yz2+2z3+6x2-24xy+4y2-23xz+12yz+30z2+17x-14z-4;
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Elimination
lexikographical ordering
β1
βn
αn
1
xα
1 · . . . · xn > x1 · . . . · xn if αj = βj for j ≤ k − 1 and αk > βk
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 34
Elimination
lexikographical ordering
β1
βn
αn
1
xα
1 · . . . · xn > x1 · . . . · xn if αj = βj for j ≤ k − 1 and αk > βk
I ⊂ R[x1 , . . . , x] ideal, G Gröbner basis, then G ∩ R[xk , . . . , xn ] is
a Gröbner basis of I ∩ R[xk , . . . , xn ].
this means geometrically to compute the projection
π : V (I) ⊂ Rn −→ Rn−k+1 .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 34
Projection
−→
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Projection
−→
π : V (z 2 − x + 1, y − xz) ⊂ C3 −→ V (x3 − x2 − y 2 ) ⊂ C2 .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 35
Projection
ring R=0,(x,y,z),dp;
ideal I=z2-x+1,y-xz;
eliminate(I,z);
_[1℄=x3-x2-y2
ring S=0,(z,x,y),lp;
ideal I=imap(R,I);
std(I);
_[1℄=x3-x2-y2
_[2℄=zy-x2+x
_[3℄=zx-y
_[4℄=z2-x+1
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 36
Blowing up a node
−→
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 37
Blowing up a node
−→
π : X = {(x, y; u : v) ∈ K 2 × P1 | xv = yu} −→ K 2 .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 37
Blowing up: SINGULAR session
LIB"resolve.lib";
ring R=0,(x,y),dp;
ideal J=y2-x2*(x+1);
ideal C=x,y;
list L=blowUp(J,C);
def S=L[1℄;
setring S;
sT;
sT[1℄=y(0)^2-x(1)-1
eD;
eD[1℄=x(1)
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Blowing up: SINGULAR session
LIB"resolve.lib";
ring R=0,(x,y),dp;
ideal J=y2-x3;
list L=resolve(J);
def S=L[1℄[1℄; setring S; showBO(BO);
==== Ambient Spae:
_[1℄=0
==== Ideal of Variety:
_[1℄=y(1)-1
==== Exeptional Divisors:
[1℄:
_[1℄=1
[2℄:
_[1℄=y(1)
[3℄:
_[1℄=x(2)
==== Images of variables of original ring:
_[1℄=x(2)^2*y(1)
_[2℄=x(2)^3*y(1)^2
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 39
Blowing up of a cusp
−→
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 40
Blowing up of a cusp
−→
π : X = {(x, y; u : v) ∈ K 2 × P1 | xv = yu} −→ K 2 .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 40
Resolution of X = V (z 2 − x2 y 2 ) ⊂ C3
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Primary decomposition
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Primary deco: SINGULAR session
LIB"primde.lib";
ring R=0,(x,y,z),dp;
ideal I=interset(ideal(y2-xz),ideal(x2,z),ideal(y,z2));
I;
I[1℄=y2z-xz2
I[2℄=x2y3-x3yz
primdeGTZ(I);
[1℄:
[2℄:
[1℄:
_[1℄=-y2+xz
[2℄:
_[1℄=-y2+xz
[3℄:
[1℄:
[1℄:
_[1℄=z2
_[1℄=z
_[2℄=y
_[2℄=x2
[2℄:
[2℄:
_[1℄=z
_[1℄=z
_[2℄=y
_[2℄=x
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 43
Primary deco: SAGE session
sage: R.<x,y,z> = QQ[℄
sage: I = (y^2-x*z)*R
sage: J = I.intersetion( (x^2,z)*R ).intersetion( (y, z^2)*R )
sage: J
Ideal (y^2*z - x*z^2, x^2*y^3 - x^3*y*z) of Multivariate Polynomial
Ring in x, y, z over Rational Field
sage: J.omplete_primary_deomposition(algorithm='gtz')
[(Ideal (-y^2 + x*z) of Multivariate Polynomial Ring in x, y, z over
Rational Field, Ideal (-y^2 + x*z) of Multivariate Polynomial Ring in
x, y, z over Rational Field), (Ideal (z^2, y) of Multivariate
Polynomial Ring in x, y, z over Rational Field, Ideal (z, y) of
Multivariate Polynomial Ring in x, y, z over Rational Field), (Ideal
(z, x^2) of Multivariate Polynomial Ring in x, y, z over Rational
Field, Ideal (z, x) of Multivariate Polynomial Ring in x, y, z over
Rational Field)℄
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 44
Primary deco: SINGULAR session
LIB"primde.lib";
ring R=0,(x,y,z),dp;
ideal I = (z2+1)^2*(z4+2)^3,(y-z3)^3,(x-yz+z4)^4;
def T = absPrimdeGTZ(I); setring T; absolute_primes;
[1℄:
[2℄:
[1℄:
[1℄:
_[1℄=a4+2
_[1℄=a2+1
_[2℄=z+a
_[2℄=z-a
_[3℄=y+a3
_[3℄=y+a
_[4℄=x
_[4℄=x
[2℄:
[2℄:
4
2
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 45
Infineon Tricore Project
Aim: prove that the processor (32 Bit) works correctly
every instruction of the processor will be verified specifying
special properties and proving them
it is difficult to check the arithmetic properties
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 46
Infineon Tricore Project
Proving arithmetic correctness of data paths in System-on-Chip
modules can be translated to the following problem:
Let I ⊂ Z/2N [x1 , . . . , xn ] be an ideal and g a polynomial. Decide
whether V (I) ⊂ V (g) ⊂ (Z/2N )n .
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 47
Note: Not every verification problem can be translated easily to
polynomials
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 48
Note: Not every verification problem can be translated easily to
polynomials
there are about 1041373247567 functions Z/232 −→ Z/232
there are about 10184 polynomial functions
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 48
Models for economy
Felix Kubler and Karl Schmedders (University of Zürich)
General problem:
Study a computer model of a national economy,
a standard exchange economy with finitely many agents and goods
especially study equilibria
Walrasian equilibrium consists of prices and choices, such that household
maximize utilities, firms maximize profits and markets clear
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 49
Models for economy
Felix Kubler and Karl Schmedders (University of Zürich)
General problem:
Study a computer model of a national economy,
a standard exchange economy with finitely many agents and goods
especially study equilibria
Walrasian equilibrium consists of prices and choices, such that household
maximize utilities, firms maximize profits and markets clear
Mathematical problem:
Find the positive real roots of a given system of polynomial
equations
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 49
equilibrium model with production
ring R = 0,x(1..22),dp;
ideal I = -1+x(1)^5*x(4)*x(13),
-1+x(3)^5*x(4)*x(15),
-1+x(2)^5*x(4)*x(14),
-1+x(5)^3*x(8)*x(13),
-1+x(7)^3*x(8)*x(15),
-1+x(6)^3*x(8)*x(14),
-1+x(9)^4*x(12)*x(13),
-1+x(10)^4*x(12)*x(14),
-1+x(11)^4*x(12)*x(15),
5+2*x(16)-x(1)*x(13)-x(2)*x(14)-x(3)*x(15),
3+5*x(16)-x(5)*x(13)-x(6)*x(14)-x(7)*x(15),
(x(1)+x(5)+x(9))^3-x(17)^2*x(18),
(x(2)+x(6)+x(10))^2-x(19)*x(20),
(x(3)+x(7)+x(11))^2-4*x(21)*x(22),
x(17)+x(19)+x(21)-10,
8*x(13)^3*x(18)-27*x(16)^3*x(17),
x(14)^2*x(20)-4*x(16)^2*x(19),
x(15)^2*x(22)-x(16)^2*x(21),
x(18)+x(20)+x(22)-10,
x(13)^3*x(17)^2-27*x(18)^2,
x(14)^2*x(19)-4*x(20),
x(15)^2*x(21)-x(22);
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 50
Solving: SINGULAR session
LIB"solve.lib";
ring R=0,(x,y,z),dp;
ideal I=(x2+1)*(x2-2),(y-1)*(y2+3),z*(z2+5);
solve(I);
[1℄:
[36℄:
[1℄:
-1.41421356
[2℄:
1
[3℄:
0
[1℄:
. . .
i
[2℄:
(i*1.73205081)
[3℄:
(i*2.236068)
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 51
Solving: SINGULAR session
LIB"solve.lib";
int i;
ring R=0,x,dp;
poly p=x+1;
for(i=2;i<=20;i++){p=p*(x+i);}
p=p+1/2^23*x19;
p;
x20+1761607681/8388608x19+20615x18+1256850x17+53327946x16+1672280820x15+40171771630x14+756111184500x13+11
list L=solve(p,1000,0,1000,"nodisplay");
def SS=L[1℄;
setring SS;
SOL;
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 52
Solving: SINGULAR session
[1℄:
-20.84690810148225691492877289263115442307494499972035663010306049580863034438969434485728258036801075
[2℄:
-8.91725024851707049429552016533513217779440514909619567438631884949595638840.....
.
.
.
[20℄:
-23.6087851262656384613468156892540659614282149346834828266821577250713081940.....
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 53
Solving: SINGULAR session
setring R;
LIB "rootsur.lib";
nrroots(p);
10
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 54
Surfaces in P3 with many singularities
m(d) := maximum number of nodes on a surface X of degree d
in P3C .
It is known: m(d) = 1, 4, 16, 31, 65 for d = 2, 3, 4, 5, 6.
5 3
d ≤ m(d) ≤ 94 d3 up to O(d2 ), but the
For d ≥ 7 we only know 12
exact value of m(d) is unknown.
Note that the lower bounds are obtained in each case by a
specific construction, due to Schläfli, Kummer, Togliatti,
Chmutov and Barth.
In 2004 O. Labs constructed a surface of degree 7 with 99 nodes
which is the current world record for surfaces of degree 7
(but which is still smaller than the known upper bound 104).
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 55
Surfaces in P3 with many singularities
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 56
Nice algebraic surface: Whitney umbrella
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 57
surfaces: http://www.imaginary2008.de/
Consider the equation
1 3 2
1
1
((y−z)2 +(y−1− 21 x− 81 x2 + 16
x ) − 50
)((x− 53 y+1)2 +(y−z 2 )2 − 100
)=0
defining a surface of degree 10 in R3 .
What do you expect from the real picture?
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 58
Nice algebraic surfaces
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 59
Nice algebraic surfaces
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 60
http://www.imaginary2008.de/
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 61
plot:Singular session
> LIB "surf.lib";
> ring R = 0, (x,y,z), dp;
> ideal I = 6/5*y^2+6/5*z^2-5*(x+1/2)^3*(1/2-x)^3;
surfer(I);
The resulting picture will show in a popup-window:
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 62
Coding theorie
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 63
Coding theory: brnoeth.lib
AG-Codes are codes, using algebraic vector spaces of divisors
on algebraic curves over finite fields
an implementation of the Brill-Noether algorithm for solving the
Riemann-Roch problem and applications in Algebraic
Geometry codes is implemented in SINGULAR.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 64
Sudoku
5
6 2
6
4
7
5 2
3 6
3
5 8
1
6
1
8
5
7
9 6
6 1
4
7
4
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 65
Sudoku
the idea of a Sudoku goes back to Leonard Euler: Latin squares
in our days invented by Howard Garns (USA): Number place
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 66
Sudoku
the idea of a Sudoku goes back to Leonard Euler: Latin squares
in our days invented by Howard Garns (USA): Number place
associate to the places in a Sudoku the variables x1 , . . . , x81
Q9
and to each variable xi the polynomial Fi (xi ) = j=1 (xi − j)
Let
E = {(i, j), i < j and i, j in the same row, column or 3 × 3 − box}
For (i, j) ∈ E let Gi,j =
Fi −Fj
xi −xj .
Let I ⊂ Q[x1 , . . . , x81 ] be the ideal generated by the 891
polynomials {Gi,j }(i,j)∈E and {Fi }i=1,...,9
a = (a1 , . . . , a81 ) ∈ V (I) iff ai ∈ {1, . . . , 9} and ai 6= aj for
(i, j) ∈ E
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 66
Sudoku
a well posed Sudoku has a unique solution.
Let L ⊂ {1, . . . , 81} be the set of pre-assigned places and
{ai }i∈L the corresponding numbers of a concrete Sudoku S.
Then IS = I+ < {xi − ai }i∈L > is the ideal associated to the
Sudoku S.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 67
Sudoku
a well posed Sudoku has a unique solution.
Let L ⊂ {1, . . . , 81} be the set of pre-assigned places and
{ai }i∈L the corresponding numbers of a concrete Sudoku S.
Then IS = I+ < {xi − ai }i∈L > is the ideal associated to the
Sudoku S.
The reduced Gröbner basis of IS with respect to the
lexicographical ordering has the shape x1 − a1 , . . . , x81 − a81
and (a1 , . . . , a81 ) is the solution of the Sudoku.
SINGULAR and Applicationstalk at theCIMPA School Lahore 2012 – p. 67