MSOLVE: Multivariate polynomial system solver

Documentation

https://msolve.lip6.fr/

msolve is an open source C library implementing computer algebra algorithms for solving polynomial systems (with rational coefficients or coefficients in a prime field).

Currently, with msolve, you can basically solve multivariate polynomial systems. This encompasses:

• the computation of Groebner bases
• real root isolation of the solutions to polynomial systems
• the computation of the dimension and the degree of the solution set and many other things you can do using msolve.

A tutorial is available here

Some of the functionalities of msolve are already available in the computer algebra systems Oscar and SageMath. See below for some more information about this.

Install Instructions

See the INSTALL file.

Input File Format

More informations are given in the tutorial (see https://msolve.lip6.fr)

msolve input files need to be of the following format:

1st line: variables as commata separated list, e.g. x1,x2,x3,x4,y1,y2.
2nd line: field characteristic, e.g. 0.
following lines: generating polynomials, all but the last one need to terminate by a ,, e.g.

x1,x2,x3,x4,y1,y2
101
x1+x2+x3+x4,
2*y1-145*y2

Polynomials may be multiline, thus , as a separator.

Coefficients can be rational, using /, e.g. -2/3*x2*y1^2+....

Basic usage

Some basic commands are as follows:

./msolve -f in.ms -o out.ms

will:

• detect if the input system has dimension at most 0
• when the system has dimension at most 0 and the coefficients are rational numbers, msolve will isolate the real solutions
• when the system has dimension at most 0 and the coefficients are in a prime field, msolve will compute a parametrization of the solutions

All output data are displayed in the file out.ms

The -vflag allows you to control the verbosity, giving insight on what msolve is doing. Try this.

./msolve -v 2 -f in.ms -o out.ms

Computing Groebner bases

msolve computes Groebner bases when the base field is either the field of rational numbers or a prime field (characteristic should be less than 2^31).

The following command

./msolve -g 1 -f in.ms -o out.ms

will compute the leading monomials of the reduced Groebner basis of the ideal generated by the input system in in.ms for the so-called graded reverse lexicographic ordering. This allows you to deduce the dimension of the solution set to the input polynomials (in an algebraic closure of the base field) as well as the degree of the ideal they generate.

Using the -g 2 flag as follows

./msolve -g 2 -f in.ms -o out.ms

will return the reduced Groebner basis for the graded reverse lexicographic ordering.

msolve also allows you to perform Groebner bases computations using one-block elimination monomial order thanks to the -e flag. The following command

./msolve -e 1 -g 2 -f in.ms -o out.ms

will perform the Groebner basis computation eliminating the first variable. More generally, using -e k will eliminate the first k variables.

Solving over the real numbers

When the input polynomial system has rational coefficients and when it has finitely many complex solutions, msolve will, by default, compute the real solutions to the input system. Those are encoded with isolating boxes for all coordinates to all real solutions.

For instance, on input file in.ms as follows

x, y
0
x^2+y^2-4,
x*y-1

the call ./msolve -f in.ms -o out.ms will display in the file out.ms the following output

[0, [1,
[[[-41011514734338452707966945920 / 2^96, -41011514734338452707966945917 / 2^96], [-153057056683910732545430822374 / 2^96, -153057056683910732545430822373 / 2^96]], 
[[-612228226735642930181723289497 / 2^98, -612228226735642930181723289492 / 2^98], [-164046058937353810831867783675 / 2^98, -164046058937353810831867783674 / 2^98]], 
[[612228226735642930181723289492 / 2^98, 612228226735642930181723289497 / 2^98], [164046058937353810831867783674 / 2^98, 164046058937353810831867783675 / 2^98]], 
[[41011514734338452707966945917 / 2^96, 41011514734338452707966945920 / 2^96], [153057056683910732545430822373 / 2^96, 153057056683910732545430822374 / 2^96]]]
]]:

which are the 4 isolating boxes of the 4 exact roots whose numerical approximations are (-0.5176380902, -1.931851653), (-1.931851653, -0.5176380902), (1.931851653, 0.5176380902) and (0.5176380902, 1.931851653).

Multi-threading

Several components of msolve are parallelized through multi-threading. Typing

./msolve -t 4 -f in.ms -o out.ms

tells msolve to use 4 threads. Multi-threading in msolve is used in

• linear algebra algorithms used for Groebner bases computations over prime fields
• multi-modular computations for solving over the reals (all intermediate and independent prime computations are run in parallel)
• algorithms for real root isolation.

msolve in AlgebraicSolving

AlgebraicSolving is a Julia package that wraps msolve and provides some more functionality like computing rational solutions. See here for more information and documentation.

msolve in Oscar

msolve is used in Oscar to solve polynomial systems with rational coefficients.

It will detect if the input system has finitely many complex solutions, in which case it will output a rational parametrization of the solution set as well as the real solutions to the input system (see msolve's tutorial here).

You can have a look at this and the documentation of Oscar.

Here is how you can use it.

julia> R,(x1,x2,x3) = PolynomialRing(QQ, ["x1","x2","x3"])
(Multivariate Polynomial Ring in x1, x2, x3 over Rational Field, fmpq_mpoly[x1, x2, x3])
julia> I = ideal(R, [x1+2*x2+2*x3-1, x1^2+2*x2^2+2*x3^2-x1, 2*x1*x2+2*x2*x3-x2])
ideal(x1 + 2*x2 + 2*x3 - 1, x1^2 - x1 + 2*x2^2 + 2*x3^2, 2*x1*x2 + 2*x2*x3 - x2)
julia> real_solutions(I)
((84*x^4 - 40*x^3 + x^2 + x, 336*x^3 - 120*x^2 + 2*x + 1, PolyElem[-184*x^3 + 80*x^2 - 4*x - 1, -36*x^3 + 18*x^2 - 2*x], fmpz[-1, -1]), Vector{fmpq}[[744483363399261433351//1180591620717411303424, 372241681699630716673//1180591620717411303424, -154187553040555781639//1180591620717411303424], [1, 0, 0], [71793683196126133110381699745//316912650057057350374175801344, 71793683196126133110381699745//633825300114114700748351602688, 173325283664805084153412401855//633825300114114700748351602688], [196765270119568550571//590295810358705651712, 1//590295810358705651712, 196765270119568550571//590295810358705651712]])

msolve in SageMath

When you have msolve installed, it is used by SageMath when you call the Variety function for solving polynomial systems with real coefficients.

You can have a look here and here

We are grateful to Marc Mezzarobba who initiated the usage of msolvein SageMath and the whole development team of SageMath, in particular those involed in this ticket

Citing msolve

If you have used msolve in the preparation of some paper, we are grateful that you cite it as follows:

msolve: A Library for Solving Polynomial Systems, 
J. Berthomieu, C. Eder, M. Safey El Din, Proceedings of the  
46th International Symposium on Symbolic and Algebraic Computation (ISSAC), 
pp. 51-58, ACM, 2021.

or, if you use BibTeX entries:

@inproceedings{msolve,
  TITLE = {{msolve: A Library for Solving Polynomial Systems}},
  AUTHOR = {Berthomieu, J{\'e}r{\'e}my and Eder, Christian and {Safey El Din}, Mohab},
  BOOKTITLE = {{2021 International Symposium on Symbolic and Algebraic Computation}},
  ADDRESS = {Saint Petersburg, Russia},
  SERIES = {46th International Symposium on Symbolic and Algebraic Computation},
  PAGES     = {51--58},
  PUBLISHER = {{ACM}},  
  YEAR = {2021},
  MONTH = Jul,
  DOI = {10.1145/3452143.3465545},
  PDF = {https://hal.sorbonne-universite.fr/hal-03191666v2/file/main.pdf},
  HAL_ID = {hal-03191666},
  HAL_VERSION = {v2},
}

The paper can be downloaded here.

Funding

The development of msolve is supported by the Forschungsinitiative Rheinland-Pfalz.