Macaulay

For more information see the Macaulay home page.

For experts, you probably want to use Macaulay from within emacs. The documentation in the above link tells you how to do that.

To start Macaulay on the mathematics computers, simply type "M2".

Here's the example I did in class:

Macaulay 2, version 0.9.2
--Copyright 1993-2001, D. R. Grayson and M. E. Stillman
--Singular-Factory 2.0.5, copyright 1993-2001, G.-M. Greuel, et al.
--Singular-Libfac 2.0.4, copyright 1996-2001, M. Messollen

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x^2-y^3,x^3-y^7)

               3    2     7    3
o2 = ideal (- y  + x , - y  + x )

o2 : Ideal of R

i3 : gb I

o3 = | y3-x2 x4y-x3 x6-x3y2 |

o3 : GroebnerBasis

i4 : x^9 % I

      4
o4 = x

o4 : R

i5 : restart
Macaulay 2, version 0.9.2
--Copyright 1993-2001, D. R. Grayson and M. E. Stillman
--Singular-Factory 2.0.5, copyright 1993-2001, G.-M. Greuel, et al.
--Singular-Libfac 2.0.4, copyright 1996-2001, M. Messollen

i1 : R = QQ[x,y,MonomialOrder=>Lex]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x^2-y^3,x^3-y^7)

             2    3   3    7
o2 = ideal (x  - y , x  - y )

o2 : Ideal of R

i3 : gb I

o3 = | x2-y3 xy3-y7 y11-y6 |

o3 : GroebnerBasis

i5 : restart
Macaulay 2, version 0.9.2
--Copyright 1993-2001, D. R. Grayson and M. E. Stillman
--Singular-Factory 2.0.5, copyright 1993-2001, G.-M. Greuel, et al.
--Singular-Libfac 2.0.4, copyright 1996-2001, M. Messollen

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : m = matrix({{x_1,x_2,x_3},{x_2,x_3,x_4}})

o2 = | x_1 x_2 x_3 |
     | x_2 x_3 x_4 |

             2       3
o2 : Matrix R  <--- R

i3 : I = minors(2,m)

               2                           2
o3 = ideal (- x  + x x , - x x  + x x , - x  + x x )
               2    1 3     2 3    1 4     3    2 4

o3 : Ideal of R

i4 : hilbertPolynomial(R/I, Projective=>false)

o4 = 3$i + 1

o4 : QQ [$i]

i5 : restart
Macaulay 2, version 0.9.2
--Copyright 1993-2001, D. R. Grayson and M. E. Stillman
--Singular-Factory 2.0.5, copyright 1993-2001, G.-M. Greuel, et al.
--Singular-Libfac 2.0.4, copyright 1996-2001, M. Messollen

i1 : R = QQ[a,b,r,x,y,MonomialOrder=>Lex]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(a^2+b^2-1,y-r*a,x-r*b,r-2*a*b)

             2    2
o2 = ideal (a  + b  - 1, - a*r + y, - b*r + x, - 2a*b + r)

o2 : Ideal of R

i3 : gb I

o3 = | x6+3x4y2+3x2y4-4x2y2+y6 ry3+1/2x5+x3y2+1/2xy4-2xy2 rxy-1/2x4-x2y2-1/2y4 rx2+ry2-2xy r2-x2-y2 by-1/2r2 
       bx2+1/2r2y-rx br-x b2x+1/2ry-x b3-b+1/2y ay+bx-r ax-by ar-y ab-1/2r a2+b2-1 |

o3 : GroebnerBasis

i4 : 

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