8.3 Controlling Groebner basis computations

One can control a Groebner basis computation by setting various parameters. These parameters can be set and examined by a built-in function dp_gr_flags(). Without argument it returns the current settings.

[100] dp_gr_flags();
[Demand,0,NoSugar,0,NoCriB,0,NoGC,0,NoMC,0,NoRA,0,NoGCD,0,Top,0,
ShowMag,1,Print,1,Stat,0,Reverse,0,InterReduce,0,Multiple,0]
[101]

The return value is a list which contains the names of parameters and their values. The meaning of the parameters are as follows. ‘on’ means that the parameter is not zero.

NoSugar

If ‘on’, Buchberger’s normal strategy is used instead of sugar strategy.

NoCriB

If ‘on’, criterion B among the Gebauer-Moeller’s criteria is not applied.

NoGC

If ‘on’, the check that a Groebner basis candidate is indeed a Groebner basis, is not executed.

NoMC

If ‘on’, the check that the resulting polynomials generates the same ideal as the ideal generated by the input, is not executed.

NoRA

If ‘on’, the interreduction, which makes the Groebner basis reduced, is not executed.

NoGCD

If ‘on’, content removals are not executed during a Groebner basis computation over a rational function field.

Top

If ‘on’, Only the head term of the polynomial being reduced is reduced.

Reverse

If ‘on’, the selection strategy of reducer in a normal form computation is such that a newer reducer is used first.

Print

If ‘on’, various information during a Groebner basis computation is displayed.

PrintShort

If ‘on’ and Print is ‘off’, short information during a Groebner basis computation is displayed.

Stat

If ‘on’, a summary of information is shown after a Groebner basis computation. Note that the summary is always shown if Print is ‘on’.

ShowMag

If ‘on’ and Print is ‘on’, the sum of bit length of coefficients of a generated basis element, which we call magnitude, is shown after every normal computation. After comleting the computation the maximal value among the sums is shown.

Content

Multiple

If a non-zero rational number, in a normal form computation over the rationals, the integer content of the polynomial being reduced is removed when its magnitude becomes Content times larger than a registered value, which is set to the magnitude of the input polynomial. After each content removal the registered value is set to the magnitude of the resulting polynomial. Content is equal to 1, the simiplification is done after every normal form computation. It is empirically known that it is often efficient to set Content to 2 for the case where large integers appear during the computation. An integer value can be set by the keyword Multiple for backward compatibility.

Demand

If the value (a character string) is a valid directory name, then generated basis elements are put in the directory and are loaded on demand during normal form computations. Each elements is saved in the binary form and its name coincides with the index internally used in the computation. These binary files are not removed automatically and one should remove them by hand.

If Print is ‘on’, the following information are shown.

[93] gr(cyclic(4),[c0,c1,c2,c3],0)$
mod= 99999989, eval = []
(0)(0)<<0,2,0,0>>(2,3),nb=2,nab=5,rp=2,sugar=2,mag=4
(0)(0)<<0,1,2,0>>(1,2),nb=3,nab=6,rp=2,sugar=3,mag=4
(0)(0)<<0,1,1,2>>(0,1),nb=4,nab=7,rp=3,sugar=4,mag=6
.
(0)(0)<<0,0,3,2>>(5,6),nb=5,nab=8,rp=2,sugar=5,mag=4
(0)(0)<<0,1,0,4>>(4,6),nb=6,nab=9,rp=3,sugar=5,mag=4
(0)(0)<<0,0,2,4>>(6,8),nb=7,nab=10,rp=4,sugar=6,mag=6
....gb done
reduceall
.......
membercheck
(0,0)(0,0)(0,0)(0,0)
gbcheck total 8 pairs
........
UP=(0,0)SP=(0,0)SPM=(0,0)NF=(0,0)NFM=(0.010002,0)ZNFM=(0.010002,0)
PZ=(0,0)NP=(0,0)MP=(0,0)RA=(0,0)MC=(0,0)GC=(0,0)T=40,B=0 M=8 F=6
D=12 ZR=5 NZR=6 Max_mag=6
[94]

In this example mod and eval indicate moduli used in trace-lifting. mod is a prime and eval is a list of integers used for evaluation when the ground field is a field of rational functions.

The following information is shown after every normal form computation.

(TNF)(TCONT)HT(INDEX),nb=NB,nab=NAB,rp=RP,sugar=S,mag=M

Meaning of each component is as follows.

TNF

CPU time for normal form computation (second)

TCONT

CPU time for content removal(second)

HT

Head term of the generated basis element

INDEX

Pair of indices which corresponds to the reduced S-polynomial

NB

Number of basis elements after removing redundancy

NAB

Number of all the basis elements

RP

Number of remaining pairs

S

Sugar of the generated basis element

M

Magnitude of the generated basis element (shown if ShowMag is ‘on’.)

The summary of the information shown after a Groebner basis computation is as follows. If a component shows timings and it contains two numbers, they are a pair of time for computation and time for garbage collection.

UP

Time to manipulate the list of critical pairs

SP

Time to compute S-polynomials over the rationals

SPM

Time to compute S-polynomials over a finite field

NF

Time to compute normal forms over the rationals

NFM

Time to compute normal forms over a finite field

ZNFM

Time for zero reductions in NFM

PZ

Time to remove integer contents

NP

Time to compute remainders for coefficients of polynomials with coefficients in the rationals

MP

Time to select pairs from which S-polynomials are computed

RA

Time to interreduce the Groebner basis candidate

MC

Time to check that each input polynomial is a member of the ideal generated by the Groebner basis candidate.

GC

Time to check that the Groebner basis candidate is a Groebner basis

T

Number of critical pairs generated

B, M, F, D

Number of critical pairs removed by using each criterion

ZR

Number of S-polynomials reduced to 0

NZR

Number of S-polynomials reduced to non-zero results

Max_mag

Maximal magnitude among all the generated polynomials

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