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gstic
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Post subject: minimal set of generators Posted: Tue Jun 22, 2021 12:22 am |
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Joined: Fri Jun 24, 2011 10:10 am Posts: 12
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minbase and mstd returns a minimal set of generators of an ideal, if the input is homogeneous or if the ordering is local Given polynomials g=x^3+y^2-1, gx=diff(g, x), gy=diff(g, y) the system of generators for the syzygies given by Singular is not minimal. Singular gives for the generators of ideal I=(gx, gy, g): syz[1]=(2xy, 3y^2-3, -6y) syz[2]=(-2y, 3x^2, 0) syz[3]=(x^3+y^2-1, 0, -3x^2) but a minimal set of generators in this case is s1=(2xy, 3y^2-3, -6y) s2=(2x^3-2, 3x^2y, -6x^2) We can easily check that syz[1], syz[2] and syz[3] can be expressed in term of s1 and s2. How can I find a minimal set of generators if the ideal is not with homogeneous polynomials and the ordering is not local ?
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hannes
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Post subject: Re: minimal set of generators Posted: Thu Jul 01, 2021 10:10 am |
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Joined: Wed May 25, 2005 4:16 pm Posts: 275
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reply by Douglas Leonard: For what it is worth, to produce your results I first used mres to get a syzygy: ring r=0,(x,y),dp; > module I=[2xy,3y2-3,-6y],[-2y,3x2,0],[x3+y2-1,0,-3x2]; > resolution mre=mres(I,0); > mre[2]; _[1]=x2*gen(1)-y2*gen(2)-2y*gen(3)+gen(2) --------------------------------------------------------- Then I rewrote the syzygy as: gen(2)=y*(y*gen(2)+2*gen(3))-x^2*gen(1) --------------------------------------------------- Then I defined: s_1:=gen(1) s_2:=y*gen(2)+2*gen(3) --------------------------------- And finally showed they generated: gen(1)=s_1 gen(2)=y*s_2-x^2*s_1 2*gen(3)=s_2-y*(y*s_2-x^2*s_1)=(1-y^2)s_2+x^2*y*s_1
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