normalI in reesclos.lib and monomial orderings

[leonada@…]

While not a bug per se, the timer values should say something about the code written:

SINGULAR /

A Computer Algebra System for Polynomial Computations / version 3-1-5

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by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann \ Jul 2012

FB Mathematik der Universitaet, D-67653 Kaiserslautern \

LIB "normal.lib"; ring r=2,(x,y),dp; ideal i=x2+xy3+x5+y5,xy4+y5+x6+xy5+y6; list nor=normalI(i);nor;

[1]:

_[1]=x5+y5+xy3+x2 _[2]=x6+xy5+y6+xy4+y5 _[3]=x4y4+x4y3+x3y3+x2y4+xy5+y5+x4+xy3+x3

timer;

45

ring r=2,(y,x),dp;

redefining r

ideal i=x2+xy3+x5+y5,xy4+y5+x6+xy5+y6; list nor=normalI(i);nor;

[1]:

_[1]=y5+x5+y3x+x2 _[2]=y6+y5x+x6+y5+y4x _[3]=y8+y4x4+y3x5+y3x4+y6+y4x+y3x

[2]:

_[1]=x17+y8x7+y8x5+y6x5+x11+y8x+y6x3+y4x5+y2x7+x7 _[2]=y10+x10+y6x2+x4 _[3]=y11+y10x+y6x5+x11+y10+y8x2+y5x5+y4x6+y3x7+y8x+y7x2+y6x2+y5x3+x8+y5x2+y4x3 _[4]=y13+y9x4+y4x9+y3x10+y11x+y8x4+y7x5+y6x6+y3x9+y11+y8x2+y3x7+y8x+y7x2+y4x3+y3x3 _[5]=y12+y10x2+x12+y10+y8x2 _[6]=y14+y13x+y10x4+y4x10+y3x11+y13+y12x+y8x5+y7x6+y3x10+y12+y11x+y8x4+y7x5+y6x6+y11+y9x2+y4x7+y3x7+y8x+y7x2 _[7]=y16+y8x8+y6x10+y6x8+y12+y8x2+y6x2

timer;

19843

[leonada@…]

Pfister said that SingularTeam? was responsible for maintatining reesclos.lib, but I have no idea how to reach SingularTeam? other than by reporting this as a bug.

[leonada]

LIB "/home/leonada/reesclos.lib"; ring r=2,(x,y),dp; ideal i=x5+x2,x4y+x2,y5+y2; list nor=normalI(i);nor;

[1]:

_[1]=x5+x2 _[2]=x4y+x2 _[3]=y5+y2 _[4]=xy4+y5+xy+y2

[2]:

_[1]=x10+x4 _[2]=x9y+x7+x6y+x4 _[3]=x5y5+x5y2+x2y5+x2y2 _[4]=x6y4+x5y5+x6y+x5y2+x3y4+x2y5+x3y+x2y2 _[5]=x8y2+x4 _[6]=x4y6+x4y3+x2y5+x2y2 _[7]=x5y5+x4y6+x5y2+x4y3+x3y4+x2y5+x3y+x2y2 _[8]=y10+y4 _[9]=xy9+y10+xy3+y4 _[10]=x2y8+y10+x2y2+y4

[3]:

_[1]=x6y6+x5y7+x6+x5y _[2]=x15+x12+x9+x6 _[3]=x14y+x12+x8y+x6 _[4]=x10y5+x10y2+x4y5+x4y2 _[5]=x11y4+x10y5+x11y+x10y2+x5y4+x4y5+x5y+x4y2 _[6]=x13y2+x10y2+x9+x6 _[7]=x9y6+x9y3+x7y5+x6y6+x7y2+x6y3+x4y5+x4y2 _[8]=x10y5+x9y6+x10y2+x9y3+x8y4+x6y6+x8y+x6y3+x5y4+x4y5+x5y+x4y2 _[9]=x5y10+x2y10+x5y4+x2y4 _[10]=x6y9+x5y10+x3y9+x2y10+x6y3+x5y4+x3y3+x2y4 _[11]=x7y8+x5y10+x4y8+x2y10+x7y2+x5y4+x4y2+x2y4 _[12]=x12y3+x10y2+x8y+x6 _[13]=x8y7+x8y4+x4y5+x4y2 _[14]=x9y6+x8y7+x9y3+x8y4+x5y4+x4y5+x5y+x4y2 _[15]=x4y11+x2y10+x4y5+x2y4 _[16]=x5y10+x4y11+x3y9+x2y10+x5y4+x4y5+x3y3+x2y4 _[17]=x6y9+x4y11+x4y8+x2y10+x6y3+x4y5+x4y2+x2y4 _[18]=y15+y12+y9+y6 _[19]=xy14+y15+xy11+y12+xy8+y9+xy5+y6 _[20]=x2y13+y15+x2y10+y12+x2y7+y9+x2y4+y6 _[21]=x3y12+x2y13+xy14+y15+x3y9+x2y10+xy11+y12+x3y6+x2y7+xy8+y9+x3y3+x2y4+xy5+y6

ideal i1=std(nor[1]);i1;

i1[1]=x3+x2y i1[2]=y5+y2 i1[3]=xy4+xy i1[4]=x2y3+x2

ideal i2=std(nor[2]);i2;

i2[1]=x6+x4y2 i2[2]=x3y4+x2y5+x3y+x2y2 i2[3]=x4y3+x2y5+x4+x2y2 i2[4]=y10+y4 i2[5]=xy9+xy3 i2[6]=x2y8+x2y2

ideal i3=std(nor[3]);i3;

i3[1]=x6y3+x4y5+x6+x4y2 i3[2]=x9+x8y+x7y2+x6y3 i3[3]=x3y9+x2y10+x3y3+x2y4 i3[4]=x4y8+x2y10+x4y2+x2y4 i3[5]=x5y7+x2y10+x6y3+x4y5+x6+x5y+x4y2+x2y4 i3[6]=y15+y12+y9+y6 i3[7]=xy14+xy11+xy8+xy5 i3[8]=x2y13+x2y10+x2y7+x2y4

ideal iii=std(i1*i2);iii;

iii[1]=x6y3+x4y5+x6+x4y2 iii[2]=x9+x8y+x7y2+x6y3 iii[3]=x3y9+x2y10+x3y3+x2y4 iii[4]=x4y8+x2y10+x4y2+x2y4 iii[5]=x5y7+x2y10+x5y+x2y4 iii[6]=y15+y12+y9+y6 iii[7]=xy14+xy11+xy8+xy5 iii[8]=x2y13+x2y10+x2y7+x2y4

ideal ii=std(i1*i1);ii;

ii[1]=x6+x4y2 ii[2]=x3y4+x2y5+x3y+x2y2 ii[3]=x4y3+x2y5+x4+x2y2 ii[4]=y10+y4 ii[5]=xy9+xy3 ii[6]=x2y8+x2y2

The answer from normalI suggests that C(i,r),C(i^{2,r), and C(i}3,r) are all non-trivial, while an std computation above shows only C(i,r) is. Perhaps that is why normalI takes so long.