SINGULAR and Applications talk at the CIMPA School Lahore 2012

Size: px

Start display at page:

Download "SINGULAR and Applications talk at the CIMPA School Lahore 2012"

Raymond Richards
5 years ago
Views:

1 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 1 SINGULAR and Applications talk at the CIMPA School Lahore 2012 Gerhard Pfister pfister@mathematik.uni-kl.de Departement of Mathematics University of Kaiserslautern

2 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 2 SINGULAR A Computer Algebra System for Polynomial Computations with special emphasize on the needs of algebraic geometry, commutative algebra, and singularity theory

3 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 2 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 Kaiserslautern

4 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 2 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 Kaiserslautern The computer is not the philosopher s stone but the philosopher s whetstone Hugo Battus, Rekenen op taal 1983

5 Twisted cubic curve f 1,...,f m C[x 1,...,x n ] V (f 1,...,f m ) = {p C n such that f i = 0 for all i} the white curve V (y x 2, xy z) is the intersection of the red surface V (xy z) and the blue surface V (y x 2 ) SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 3

6 Cusp SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 4

7 Cup SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 5

8 A1 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 6

9 Cayley-cubic SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 7

10 Heart SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 8

11 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 9 Adding machine 1623 Wilhelm Schickard (Universität Tübingen)

12 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 10 Slide rule Let us compute 1, 3 4, 2.

13 Adding machine SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 11

14 log table SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 12

15 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 13 Logarithm Let us compute 3 5 :

16 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 13 Logarithm Let us compute 3 5 : ( log 3 5 ) = 1 3 log(5) log(5) = 0, 6990

17 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 13 Logarithm Let us compute 3 5 : ( log 3 5 ) = 1 3 log(5) log(5) = 0, log(5) = 0, 2330 = log(1, 71)

18 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 13 Logarithm Let us compute 3 5 : ( log 3 5 ) = 1 3 log(5) log(5) = 0, log(5) = 0, 2330 = log(1, 71) 1, 71 3 = 5, 0002

19 Punch card SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 14

20 ZXSpectrum SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 15

21 K1630 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 16

22 Atari SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 17

23 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 18 Birth of SINGULAR 1984

24 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

25 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?

26 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).

27 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 20 History 2004 Jenks Price for: Excellence in Software Engineering awarded at ISSAC in Santander

28 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 20 History 2004 Jenks Price for: Excellence in Software Engineering awarded at ISSAC in Santander

SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 20 History 2004 Jenks Price for: Excellence in Software Engineering awarded at ISSAC in Santander http://www.

29 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 20 History 2004 Jenks Price for: Excellence in Software Engineering awarded at ISSAC in Santander Supported by: Deutsche Forschungsgemeinschaft, Stiftung Rheinland-Pfalz für Innovation, Volkswagen Stiftung SINGULAR is free software (Gnu Public Licence)

30 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 21 Team G. Pfister,H. Schönemann, W. Decker,G.-M. Greuel

31 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 21 Team Kaiserslautern Saarbrücken Cottbus Berlin Mainz Dortmund Valladolid La Laguna Buenos Aires Cape Town... G. Pfister,H. Schönemann, W. Decker,G.-M. Greuel

32 Teams SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 22

33 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 23 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.

34 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 23 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.

35 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 24 Poincaré complex Let (X l,k, 0) be the germ of the unimodal space curve singularity FT k,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.

36 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 24 Poincaré complex Let (X l,k, 0) be the germ of the unimodal space curve singularity FT k,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 O Xl,k,0 Ω 1 X l,k,0 Ω2 X l,k,0 Ω3 X l,k,0 0 is exact. But (X l,k, 0) is not quasi-homogeneous: µ(x, 0) = τ(x, 0) + 1 = k + l + 2.

37 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) = dim C (Ω 1 X,0 /do (X,0))

38 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) = dim C (Ω 1 X,0 /do (X,0)) τ(x, 0) = dim C (C{x, y, z}/ < f, g, M 1, M 2, M 3 >) here the M i are the 2-minors of the Jacobian matrix of f, g.

39 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) = dim C (Ω 1 X,0 /do (X,0)) τ(x, 0) = dim C (C{x, y, z}/ < f, g, M 1, M 2, M 3 >) here the M i are the 2-minors of the Jacobian matrix of f, g. Reiffen: The Poincaré complex is exact if and only if < f, g > Ω 3 C 3,0 d(< f, g > Ω2 C 3,0 ) and µ(x, 0) = dim C (Ω 2 X,0 ) dim C(Ω 3 X,0 )

40 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 26 Poincaré complex Let A = C[x 1,...,x n ] or the localization of the polynomial ring in the ideal < x 1,...,x n > Let M A m be a sub-module. We need to compute the C-vectorspace dimension of A m /M.

41 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 26 Poincaré complex Let A = C[x 1,...,x n ] or the localization of the polynomial ring in the ideal < x 1,...,x n > Let M A m be a sub-module. We need to compute the C-vectorspace dimension of A m /M. dim C (C[x, y]/ < y 3, x 2 y, x 4 >= 8, because 1, x, y, x 2, xy, y 2, x 3, xy 2 is a basis. dim C (C[x, y]/ < y 3 + x 2, y 4 + xy 2 >= 8 is not so easy to see. We need Gröbner bases to see this.

42 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 27 Poincaré complex: SINGULAR session Ö Ò Ê ¼ Ü Ýµ Ô Ð Á Ý Ü¾ Ý ÜÝ¾ Ð Â Ø Áµ Ú Ñ Âµ Â Â ½ Ý Ü¾ Â ¾ Ü¾Ý¹ÜÝ¾ Â Ü Ü

43 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 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.

44 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 29 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.

45 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

46 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 31 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.

47 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 32 Elimination Marlis -Problem: Find the equations of the curve given by the parametrization: x = t 5 + t 4 + t 3 2t y = t 3 t z = t 5 + t 3

48 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 33 Elimination Ö Ò Ö ¼ Ü Ý Þ Øµ Ô Ð Á Ü¹ ½¹¾Ø¾ Ø Ø ¹Ø µ Ý¹ ½¹Ø¾¹Ø µ Þ¹ Ø ¹Ø µ Ð Â Ð Ñ Ò Ø Á Øµ Â Â ½ Ü¾Þ¹ÜÝÞ ¾Ý¾Þ¹¾ÜÞ¾ Þ Ü¾¹ ÜÝ Ý¾¹½½ÜÞ ÝÞ ½¾Þ¾ Ü¹ Þ¹½ Â ¾ Ý Ü¾¹ ÜÝ¹ ÜÞ ÝÞ ¾Þ¾ Ü¹Ý¹ Þ Â ÜÝ¾¹Ý¾Þ¹Ü¾ ÜÞ¹ Þ¾ Â Ü¾Ý¹¾ÜÝÞ¹Ý¾Þ ÝÞ¾¹ Ü¾ ÜÝ ½¼ÜÞ¹ ÝÞ¹ Þ¾¹Ü ¾Þ Â Ü ¹ ÜÝÞ Ý¾Þ¹ ÜÞ¾ ÝÞ¾ ¾Þ Ü¾¹¾ ÜÝ Ý¾¹¾ ÜÞ ½¾ÝÞ ¼Þ¾ ½ Ü¹½ Þ¹

49 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 34 Elimination lexikographical ordering x α x α n n > x β xβ n n if α j = β j for j k 1 and α k > β k

50 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 34 Elimination lexikographical ordering x α x α n n > x β xβ n n if α j = β j for j k 1 and α k > β k I R[x 1,...,x] ideal, G Gröbner basis, then G R[x k,...,x n ] is a Gröbner basis of I R[x k,...,x n ]. this means geometrically to compute the projection π : V (I) R n R n k+1.

51 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 35 Projection

52 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 35 Projection π : V (z 2 x + 1, y xz) C 3 V (x 3 x 2 y 2 ) C 2.

53 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 36 Projection Ö Ò Ê ¼ Ü Ý Þµ Ô Ð Á Þ¾¹Ü ½ Ý¹ÜÞ Ð Ñ Ò Ø Á Þµ ½ Ü ¹Ü¾¹Ý¾ Ö Ò Ë ¼ Þ Ü Ýµ ÐÔ Ð Á Ñ Ô Ê Áµ Ø Áµ ½ Ü ¹Ü¾¹Ý¾ ¾ ÞÝ¹Ü¾ Ü ÞÜ¹Ý Þ¾¹Ü ½

54 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 37 Blowing up a node

55 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 37 Blowing up a node π : X = {(x, y; u : v) K 2 P 1 xv = yu} K 2.

56 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 38 Blowing up: SINGULAR session ÄÁ Ö ÓÐÚ ºÐ Ö Ò Ê ¼ Ü Ýµ Ô Ð Â Ý¾¹Ü¾ Ü ½µ Ð Ü Ý Ð Ø Ä ÐÓÛÍÔ Â µ Ë Ä ½ ØÖ Ò Ë Ì Ì ½ Ý ¼µ ¾¹Ü ½µ¹½ ½ Ü ½µ

57 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 39 Blowing up: SINGULAR session ÄÁ Ö ÓÐÚ ºÐ Ö Ò Ê ¼ Ü Ýµ Ô Ð Â Ý¾¹Ü Ð Ø Ä Ö ÓÐÚ Âµ Ë Ä ½ ½ ØÖ Ò Ë ÓÛ Ç Çµ Ñ ÒØ ËÔ ½ ¼ Á Ð Ó Î Ö ØÝ ½ Ý ½µ¹½ Ü ÔØ ÓÒ Ð Ú ÓÖ ½ ¾ ½ ½ ½ Ý ½µ ½ Ü ¾µ ÁÑ Ó Ú Ö Ð Ó ÓÖ Ò Ð Ö Ò ½ Ü ¾µ ¾ Ý ½µ ¾ Ü ¾µ Ý ½µ ¾

58 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 40 Blowing up of a cusp

59 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 40 Blowing up of a cusp π : X = {(x, y; u : v) K 2 P 1 xv = yu} K 2.

60 Resolution of X = V (z 2 x 2 y 2 ) C 3 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 41

61 Primary decomposition SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 42

62 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 43 Primary deco: SINGULAR session ÄÁ ÔÖ Ñ ºÐ Ö Ò Ê ¼ Ü Ý Þµ Ô Ð Á ÒØ Ö Ø Ð Ý¾¹ÜÞµ Ð Ü¾ Þµ Ð Ý Þ¾µµ Á Á ½ Ý¾Þ¹ÜÞ¾ Á ¾ Ü¾Ý ¹Ü ÝÞ ÔÖ Ñ Ì Áµ ½ ¾ ½ ½ ½ ½ ¹Ý¾ ÜÞ ½ Þ¾ ½ Þ ¾ ¾ Ý ¾ Ü¾ ½ ¹Ý¾ ÜÞ ¾ ¾ ½ Þ ½ Þ ¾ Ý ¾ Ü

63 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 44 Primary deco: SAGE session Êº Ü Ý Þ ÉÉ Á Ý ¾¹Ü Þµ Ê Â Áº ÒØ Ö Ø ÓÒ Ü ¾ Þµ Ê µº ÒØ Ö Ø ÓÒ Ý Þ ¾µ Ê µ Â Á Ð Ý ¾ Þ ¹ Ü Þ ¾ Ü ¾ Ý ¹ Ü Ý Þµ Ó ÅÙÐØ Ú Ö Ø ÈÓÐÝÒÓÑ Ð Ê Ò Ò Ü Ý Þ ÓÚ Ö Ê Ø ÓÒ Ð Ð ÂºÓÑÔÐ Ø ÔÖ Ñ ÖÝ ÓÑÔÓ Ø ÓÒ Ð ÓÖ Ø Ñ ³ ØÞ³µ Á Ð ¹Ý ¾ Ü Þµ Ó ÅÙÐØ Ú Ö Ø ÈÓÐÝÒÓÑ Ð Ê Ò Ò Ü Ý Þ ÓÚ Ö Ê Ø ÓÒ Ð Ð Á Ð ¹Ý ¾ Ü Þµ Ó ÅÙÐØ Ú Ö Ø ÈÓÐÝÒÓÑ Ð Ê Ò Ò Ü Ý Þ ÓÚ Ö Ê Ø ÓÒ Ð Ð µ Á Ð Þ ¾ Ýµ Ó ÅÙÐØ Ú Ö Ø ÈÓÐÝÒÓÑ Ð Ê Ò Ò Ü Ý Þ ÓÚ Ö Ê Ø ÓÒ Ð Ð Á Ð Þ Ýµ Ó ÅÙÐØ Ú Ö Ø ÈÓÐÝÒÓÑ Ð Ê Ò Ò Ü Ý Þ ÓÚ Ö Ê Ø ÓÒ Ð Ð µ Á Ð Þ Ü ¾µ Ó ÅÙÐØ Ú Ö Ø ÈÓÐÝÒÓÑ Ð Ê Ò Ò Ü Ý Þ ÓÚ Ö Ê Ø ÓÒ Ð Ð Á Ð Þ Üµ Ó ÅÙÐØ Ú Ö Ø ÈÓÐÝÒÓÑ Ð Ê Ò Ò Ü Ý Þ ÓÚ Ö Ê Ø ÓÒ Ð Ð µ

64 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 45 Primary deco: SINGULAR session ÄÁ ÔÖ Ñ ºÐ Ö Ò Ê ¼ Ü Ý Þµ Ô Ð Á Þ¾ ½µ ¾ Þ ¾µ Ý¹Þ µ Ü¹ÝÞ Þ µ Ì ÈÖ Ñ Ì Áµ ØÖ Ò Ì ÓÐÙØ ÔÖ Ñ ½ ¾ ½ ½ ½ ¾ ½ ¾ ½ ¾ Þ ¾ Þ¹ Ý Ý Ü Ü ¾ ¾ ¾

65 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 46 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

66 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 47 Infineon Tricore Project Proving arithmetic correctness of data paths in System-on-Chip modules can be translated to the following problem: Let I Z/2 N [x 1,...,x n ] be an ideal and g a polynomial. Decide whether V (I) V (g) (Z/2 N ) n.

67 Note: Not every verification problem can be translated easily to polynomials SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 48

68 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 48 Note: Not every verification problem can be translated easily to polynomials there are about functions Z/2 32 Z/2 32 there are about polynomial functions

69 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

70 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

71 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 50 equilibrium model with production Ö Ò Ê ¼ Ü ½ºº¾¾µ Ô Ð Á ¹½ Ü ½µ Ü µ Ü ½ µ ¹½ Ü ¾µ Ü µ Ü ½ µ ¹½ Ü µ Ü µ Ü ½ µ ¹½ Ü µ Ü µ Ü ½ µ ¹½ Ü µ Ü µ Ü ½ µ ¹½ Ü µ Ü µ Ü ½ µ ¹½ Ü µ Ü ½¾µ Ü ½ µ ¹½ Ü ½¼µ Ü ½¾µ Ü ½ µ ¹½ Ü ½½µ Ü ½¾µ Ü ½ µ ¾ Ü ½ µ¹ü ½µ Ü ½ µ¹ü ¾µ Ü ½ µ¹ü µ Ü ½ µ Ü ½ µ¹ü µ Ü ½ µ¹ü µ Ü ½ µ¹ü µ Ü ½ µ Ü ½µ Ü µ Ü µµ ¹Ü ½ µ ¾ Ü ½ µ Ü ¾µ Ü µ Ü ½¼µµ ¾¹Ü ½ µ Ü ¾¼µ Ü µ Ü µ Ü ½½µµ ¾¹ Ü ¾½µ Ü ¾¾µ Ü ½ µ Ü ½ µ Ü ¾½µ¹½¼ Ü ½ µ Ü ¾¼µ Ü ¾¾µ¹½¼ Ü ½ µ Ü ½ µ¹¾ Ü ½ µ Ü ½ µ Ü ½ µ Ü ½ µ ¾¹¾ Ü ½ µ ¾ Ü ½ µ ¾ Ü ¾¼µ¹ Ü ½ µ ¾ Ü ½ µ Ü ½ µ ¾ Ü ½ µ¹ Ü ¾¼µ Ü ½ µ ¾ Ü ¾¾µ¹Ü ½ µ ¾ Ü ¾½µ Ü ½ µ ¾ Ü ¾½µ¹Ü ¾¾µ

72 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 51 Solving: SINGULAR session ÄÁ ÓÐÚ ºÐ Ö Ò Ê ¼ Ü Ý Þµ Ô Ð Á Ü¾ ½µ Ü¾¹¾µ Ý¹½µ Ý¾ µ Þ Þ¾ µ ÓÐÚ Áµ ½ ½ ½ ¹½º ½ ¾½ º º º ¾ ¾ ½ ½º ¾¼ ¼ ½µ ¼ ¾º¾ ¼ µ

73 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 52 Solving: SINGULAR session ÄÁ ÓÐÚ ºÐ ÒØ Ö Ò Ê ¼ Ü Ô ÔÓÐÝ Ô Ü ½ ÓÖ ¾ ¾¼ µßô Ô Ü µ Ð Ô Ô ½»¾ ¾ Ü½ Ô Ü¾¼ ½ ½ ¼ ½» ¼ Ü½ ¾¼ ½ Ü½ ½¾ ¼Ü½ ¾ Ü½ ½ ¾¾ ¼ ¾¼Ü½ ¼½ ½ ½ ¼Ü½ ½½½½ ¼¼Ü Ð Ø Ä ÓÐÚ Ô ½¼¼¼ ¼ ½¼¼¼ ÒÓ ÔÐ Ý µ ËË Ä ½ ØÖ Ò ËË ËÇÄ

74 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 53 Solving: SINGULAR session ½ ¹¾¼º ¼ ½¼½ ¾¾ ½ ¾ ¾ ¾ ½½ ¾ ¼ ¾¼ ¼½¼ ¼ ¼ ¼ ¼ ¾ ¾ ¼ ¾ ¹ º ½ ¾ ¼¾ ½ ¼ ¼ ¾ ¾¼½ ½ ¾½ ¼ ½ ¼ ½ ½ ¼ººººº º º º ¾¼ ¹¾ º ¼ ½¾ ¾ ½ ½ ¾ ¼ ½ ¾ ¾½ ¾ ¾ ¾½ ¾ ¼ ½ ¼ ½ ¼ººººº

75 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 54 Solving: SINGULAR session ØÖ Ò Ê ÄÁ ÖÓÓØ ÙÖºÐ ÒÖÖÓÓØ Ôµ ½¼

76 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 55 Surfaces in P 3 with many singularities m(d) := maximum number of nodes on a surface X of degree d in P 3 C. It is known: m(d) = 1, 4, 16, 31, 65 for d = 2, 3, 4, 5, 6. For d 7 we only know 5 12 d3 m(d) 4 9 d3 up to O(d 2 ), but the 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).

77 Surfaces in P 3 with many singularities SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 56

78 Nice algebraic surface: Whitney umbrella SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 57

79 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 58 surfaces: Consider the equation ((y z) 2 +(y x 1 8 x x3 ) )((x 3 5 y+1)2 +(y z 2 ) ) = 0 defining a surface of degree 10 in R 3. What do you expect from the real picture?

80 Nice algebraic surfaces SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 59

81 Nice algebraic surfaces SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 60

82 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 61

83 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 62 plot:singular session ÄÁ ÙÖ ºÐ Ö Ò Ê ¼ Ü Ý Þµ Ô Ð Á» Ý ¾» Þ ¾¹ Ü ½»¾µ ½»¾¹Üµ ÙÖ Ö Áµ The resulting picture will show in a popup-window:

84 Coding theorie SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 63

85 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 64 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.

86 SINGULAR and Applicationstalk at thecimpa School Lahore 2012 p. 65 Sudoku

87 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

88 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 x 1,...,x 81 and to each variable x i the polynomial F i (x i ) = 9 j=1 (x i j) Let E = {(i, j), i < j and i, j in the same row, column or 3 3 box} For (i, j) E let G i,j = F i F j x i x j. Let I Q[x 1,...,x 81 ] be the ideal generated by the 891 polynomials {G i,j } (i,j) E and {F i } i=1,...,9 a = (a 1,...,a 81 ) V (I) iff a i {1,...,9} and a i a j for (i, j) E

89 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 {a i } i L the corresponding numbers of a concrete Sudoku S. Then I S = I+ < {x i a i } i L > is the ideal associated to the Sudoku S.

90 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 {a i } i L the corresponding numbers of a concrete Sudoku S. Then I S = I+ < {x i a i } i L > is the ideal associated to the Sudoku S. The reduced Gröbner basis of I S with respect to the lexicographical ordering has the shape x 1 a 1,...,x 81 a 81 and (a 1,...,a 81 ) is the solution of the Sudoku.

SINGULAR. SINGULAR and Applications SINGULAR SINGULAR

SINGULAR. SINGULAR and Applications SINGULAR SINGULAR SINGULAR SINGULAR and Applications A Computer Algebra System for Polynomial Computations with special emphasize on the needs of algebraic geometry, commutative algebra, and singularity theory G.-M. Greuel,