# Singular

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 Font size: Tiny Small Normal Large Huge Font colour [quote="AG"]We know that if \$I\$ is a graded ideal in a polynomial ring, then the graded Betti numbers of \$I\$ are at most the corresponding graded Betti numbers of the initial ideal of \$I\$ with respect to any monomial order (See Corollary 3.3.3 of the book by Herzog and Hibi). The following code in Singular says differently. Where is my mistake? [code] > ring r = 0, (x, y,z, u,v, w, a, b, c, d), Dp; > ideal P = zw+u2+uv, ya+zu+uv, xb+y2+yz+zu, uvcd+wac+wb2, zvcd-uac-ub2-vac-vb2; > P; P[1]=zw+u2+uv P[2]=ya+zu+uv P[3]=xb+y2+yz+zu P[4]=uvcd+wac+wb2 P[5]=zvcd-uac-ub2-vac-vb2 > P = std(P); > ideal P' = lead(P); > P'; P'[1]=zw P'[2]=ya P'[3]=xb P'[4]=uvcd P'[5]=zvcd > resolution R = mres(P, 0); > resolution R' = mres(P', 0); > print(betti(R), "betti"); 0 1 2 3 4 ------------------------------------ 0: 1 - - - - 1: - 3 - - - 2: - - 4 - - 3: - 2 1 3 - 4: - - 4 2 1 5: - - - 2 1 ------------------------------------ total: 1 5 9 7 2 > print(betti(R'), "betti"); 0 1 2 3 4 ------------------------------------ 0: 1 - - - - 1: - 3 - - - 2: - - 3 - - 3: - 2 2 1 - 4: - - 4 4 - 5: - - - 2 2 ------------------------------------ total: 1 5 9 7 2 [/code][/quote]
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Topic review - Possible bug in Betti diagram
Author Message
 hannes
 Post subject: Re: Possible bug in Betti diagram
 Posted: Sat Sep 23, 2017 1:09 pm
 AG
 Post subject: Possible bug in Betti diagram
 We know that if \$I\$ is a graded ideal in a polynomial ring, then the graded Betti numbers of \$I\$ are at most the corresponding graded Betti numbers of the initial ideal of \$I\$ with respect to any monomial order (See Corollary 3.3.3 of the book by Herzog and Hibi).The following code in Singular says differently. Where is my mistake? Code:>  ring r = 0, (x, y,z, u,v, w, a, b, c, d), Dp;> ideal P = zw+u2+uv, ya+zu+uv, xb+y2+yz+zu, uvcd+wac+wb2, zvcd-uac-ub2-vac-vb2; > P;P[1]=zw+u2+uvP[2]=ya+zu+uvP[3]=xb+y2+yz+zuP[4]=uvcd+wac+wb2P[5]=zvcd-uac-ub2-vac-vb2> P = std(P);> ideal P' = lead(P);> P';P'[1]=zwP'[2]=yaP'[3]=xbP'[4]=uvcdP'[5]=zvcd> resolution R = mres(P, 0);> resolution R' = mres(P', 0);> print(betti(R), "betti");           0     1     2     3     4------------------------------------    0:     1     -     -     -     -    1:     -     3     -     -     -    2:     -     -     4     -     -    3:     -     2     1      3     -    4:     -     -     4      2     1    5:     -     -     -       2     1------------------------------------total:     1     5     9     7     2> print(betti(R'), "betti");           0     1     2     3     4------------------------------------    0:     1     -     -     -     -    1:     -     3     -     -     -    2:     -     -     3     -     -    3:     -     2     2     1     -    4:     -     -     4     4     -    5:     -     -     -      2     2------------------------------------total:     1     5     9     7     2 We know that if \$I\$ is a graded ideal in a polynomial ring, then the graded Betti numbers of \$I\$ are at most the corresponding graded Betti numbers of the initial ideal of \$I\$ with respect to any monomial order (See Corollary 3.3.3 of the book by Herzog and Hibi).The following code in Singular says differently. Where is my mistake? [code]> ring r = 0, (x, y,z, u,v, w, a, b, c, d), Dp;> ideal P = zw+u2+uv, ya+zu+uv, xb+y2+yz+zu, uvcd+wac+wb2, zvcd-uac-ub2-vac-vb2; > P;P[1]=zw+u2+uvP[2]=ya+zu+uvP[3]=xb+y2+yz+zuP[4]=uvcd+wac+wb2P[5]=zvcd-uac-ub2-vac-vb2> P = std(P);> ideal P' = lead(P);> P';P'[1]=zwP'[2]=yaP'[3]=xbP'[4]=uvcdP'[5]=zvcd> resolution R = mres(P, 0);> resolution R' = mres(P', 0);> print(betti(R), "betti"); 0 1 2 3 4------------------------------------ 0: 1 - - - - 1: - 3 - - - 2: - - 4 - - 3: - 2 1 3 - 4: - - 4 2 1 5: - - - 2 1------------------------------------total: 1 5 9 7 2> print(betti(R'), "betti"); 0 1 2 3 4------------------------------------ 0: 1 - - - - 1: - 3 - - - 2: - - 3 - - 3: - 2 2 1 - 4: - - 4 4 - 5: - - - 2 2------------------------------------total: 1 5 9 7 2[/code]
 Posted: Fri Sep 22, 2017 3:26 pm

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