System of polynomial equations

A system of polynomial equations is a set of simultaneous equations f₁ = 0, ..., f_h = 0 where the f_i are polynomials in several variables, say x₁, ..., x_n, over some field k.

Usually, the field k is either the field of rational numbers or a finite field, although most of the theory applies to any field.

A solution is a set of the values for the x_i which make all of the equations true and which belong to some algebraically closed field extension K of k. When k is the field of rational numbers, K is the field of complex numbers.

Examples and extensions
Trigonometric equations

A trigonometric equation is an equation g = 0 where g is a trigonometric polynomial. Such an equation may be converted into a polynomial system by expanding the sines and cosines in it, replacing sin(x) and cos(x) by two new variables s and c and adding the new equation s² + c² − 1 = 0.

For example the equation

\( \sin(x)^3+\cos(3x)=0 \, \)

is equivalent to the polynomial system

\( \begin{cases} s^3+4c^3-3c&=0\\ s^2+c^2-1&=0 \end{cases} \)

Solutions in a finite field

When solving a system over a finite field k with q elements, one is primarily interested in the solutions in k. As the elements of k are exactly the solutions of the equation x^q − x = 0, it suffices, for restricting the solutions to k, to add the equation x_i^q − x_i = 0 for each variable x_i.

Coefficients in a number field or in a finite field with non-prime order

The elements of a number field are usually represented as polynomials in a generator of the field which satisfies some univariate polynomial equation. To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add the equation of the generator to the equations of the system. Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers.

For example, if a system contains \( \sqrt{2} \), a system over the rational numbers is obtained by adding the equation r₂² − 2 = 0 and replacing \( \sqrt{2} \) by r₂ in the other equations.

In the case of a finite field, the same transformation allows always to suppose that the field k has a prime order.

Basic properties and definitions

A system is overdetermined if the number of equations is higher than the number of variables. A system is inconsistent if it has no solutions. By Hilbert's Nullstellensatz this means that 1 is a linear combination (with polynomials as coefficients) of the first members of the equations. Most but not all overdetermined systems are inconsistent. For example the system  x³ − 1 = 0, x² − 1 = 0 is overdetermined (having two equations but only one unknown), but it is not inconsistent since it has the solution x =1.

A system is underdetermined if the number of equations is lower than the number of the variables. An underdetermined system is either inconsistent or has infinitely many solutions in an algebraically closed extension K of k.

A system is zero-dimensional if it has a finite number of solutions in an algebraically closed extension K of k. This terminology comes from the fact that the algebraic variety of the solutions has dimension zero. A system with infinitely many solutions is said to be positive-dimensional.

A zero-dimensional system with as many equations as variables is said to be well-behaved.^[1] Bézout's theorem asserts that a well-behaved system whose equations have degrees d₁, ..., d_n has at most d₁...d_n solutions. This bound is sharp. If all the degrees are equal to d, this bound becomes dⁿ and is exponential in the number of variables.

This exponential behavior makes solving polynomial systems difficult and explains why there are few solvers that are able to automatically solve systems with Bézout's bound higher than, say, 25 (three equations of degree 3 or five equations of degree 2 are beyond this bound).

What is solving?

The first thing to do for solving a polynomial system is to decide if it is inconsistent, zero-dimensional or positive dimensional. This may be done by the computation of a Gröbner basis of the left-hand sides of the equations. The system is inconsistent if this Gröbner basis is reduced to 1. The system is zero-dimensional if, for every variable there is a leading monomial of some element of the Gröbner basis which is a pure power of this variable. For this test, the best monomial order is usually the graded reverse lexicographic one (grevlex).

If the system is positive-dimensional, it has infinitely many solutions. It is thus not possible to enumerate them. It follows that, in this case, solving may only mean "finding a description of the solutions from which the relevant properties of the solutions are easy to extract". There is no commonly accepted such description. In fact there are many different "relevant properties", which involve almost every subfield of algebraic geometry.

A natural example of an open question about solving positive-dimensional systems is the following: decide if a polynomial system over the rational numbers has a finite number of real solutions and compute them. The only published algorithm which allows one to solve this question is cylindrical algebraic decomposition, which is not efficient enough, in practice, to be used for this.

For zero-dimensional systems, solving consists of computing all the solutions. There are two different ways of outputting the solutions. The most common, possible only for real or complex solutions consists in outputting numeric approximations of the solutions. Such a solution is called numeric. A solution is certified if it is provided with a bound on the error of the approximations which separates the different solutions.

The other way to represent the solutions is said to be algebraic. It uses the fact that, for a zero-dimensional system, the solutions belong to the algebraic closure of the field k of the coefficients of the system. There are several ways to represent the solution in an algebraic closure, which are discussed below. All of them allow one to compute a numerical approximation of the solutions by solving one or several univariate equations. For this computation, the representation of the solutions which need only to solve only one univariate polynomial for each solution have to be preferred: computing the roots of a polynomial which has approximate coefficients is a highly unstable problem.

Algebraic representation of the solutions
Regular chains
Main article: Regular chain

The usual way of representing the solutions is through zero-dimensional regular chains. Such a chain consists in a sequence of polynomials f₁(x₁), f₂(x₁, x₂), ..., f_n(x₁, ..., x_n) such that, for every i such that 1 ≤ i ≤ n

• f_i is a polynomial in x₁, ..., x_i only, which has a degree d_i > 0 in x_i;
• the coefficient of x_i^d_i in f_i is a polynomial in x₁, ..., x_{i −1} which does not have any common zero with f₁, ..., f_{i − 1}.

To such a regular chain is associated a triangular system of equations

\( \begin{cases} f_1(x_1)= 0\\ f_2(x_1,x_2)=0\\ \cdots\\ f_n(x_1, x_2, \ldots, x_n)=0. \end{cases} \)

The solutions of this system are obtained by solving the first univariate equation, substituting the solutions in the other equations, then solving the second equation which is now univariate, and so on. The definition of regular chains implies that the univariate equation obtained from f_i has degree d_i and thus that the system has d₁ ... d_n solutions, provided that there is no multiple root in this resolution process (fundamental theorem of algebra).

Every zero-dimensional system of polynomial equations is equivalent (i.e. has the same solutions) to a finite number of regular chains. Several regular chains may be needed, as it is the case for the following system which has three solutions.

\( \begin{cases} x^2-1=0\\ (x-1)(y-1)=0\\ y^2-1=0 \end{cases} \)

There are several algorithms for computing a triangular decomposition of an arbitrary polynomial system (not necessarily zero-dimensional)[2] into regular chains (or regular semi-algebraic systems).

There is also an algorithm which is specific to the zero-dimensional case and is competitive, in this case, with the direct algorithms. It consists in computing first the Gröbner basis for the graded reverse lexicographic order (grevlex), then deducing the Gröbner basis by FGLM algorithm[3] and finally applying the Lextriangular algorithm.[4]

This representation of the solutions are fully convenient for coefficients in a finite field. However, for rational coefficients, There are two problems:

The output uses to involve huge integers which may make the computation and the use of the result problematic.
To deduce the numeric values of the solutions from the output, one has to solve univariate polynomials with approximate coefficients, which is a highly unstable problem.

The first issue has been theoretically solved by Dahan and Schost:[5][6] Among the sets of regular chains that represent a given set of solutions, there is a set for which the coefficients are explicitly bounded in terms of the size of the input system, with a nearly optimal bound. This set, called equiprojectable decomposition, depends only on the choice of the coordinates. This allows to use modular methods for computing efficiently the equiprojectable decomposition.[7]

The second issue is generally solved by outputting regular chains of a special form, sometimes called shape lemma, for which all di but the first one are equal to 1. For getting such regular chains, one may have to add a further variable, called separating variable, which is given the index 0. The rational univariate representation, described below, allows to compute such a special regular chain, satisfying Dahan–Schost bound, by starting from either a regular chain or a Gröbner basis.

Rational Univariate Representation

The rational univariate representation or RUR is a representation of the solutions of a zero-dimensional polynomial system over the rational numbers which has been introduced by F. Rouillier [8] for remedying to the above drawbacks of the regular chain representation.

A RUR of a zero-dimensional system consists in a linear combination x0 of the variables, called separating variable, and a system of equations[9]

\( \begin{cases} h(x_0)=0\\ x_1=g_1(x_0)/g_0(x_0)\\ \cdots\\ x_n=g_n(x_0)/g_0(x_0) \end{cases} \)

where h is a univariate polynomial in x₀ of degree D and g₀, ..., g_n are univariate polynomials in x₀ of degree less than D.

Given a zero-dimensional polynomial system over the rational numbers, the RUR has the following properties.

All but a finite number linear combinations of the variables are separating variables.
When the separating variable is chosen, the RUR exists and is unique. In particular h and the gi are defined independently of any algorithm to compute them.
The solutions of the system are in one to one correspondence with the roots of h and the multiplicity of each root of h equals the multiplicity of the corresponding solution.
The solutions of the system are obtained by substituting the roots of h in the other equations.
If h does not have any multiple root then g0 is the derivative of h.

For example, for above system, every linear combination of the variable, except the multiples of x, y and x + y, is a separating variable. If one choose t = x – y/2 as a separating variable, then the RUR is

\( \begin{cases} t^3-t=0\\ x=\frac{t^2+2t-1}{3t^2-1}\\ y=\frac{t^2-2t-1}{3t^2-1}\\ \end{cases} \)

The RUR is uniquely defined for a given separating variable, independently of any algorithm and, it preserves the multiplicities of the roots. This is a notable difference with triangular decompositions (even the equiprojectable decomposition), which, in general, do not preserve multiplicities. The RUR shares with equiprojectable decomposition the property of producing an output with coefficients of relatively small size.

For zero-dimensional systems, the RUR allows to get the numeric values of the solutions by solving a single univariate polynomial and substituting them in rational functions. This allows to provide certified approximations of the solutions to any given precision.

Moreover, the univariate polynomial h(x0) of the RUR may be factorized, and this gives a RUR for every irreducible factor. This provides the prime decomposition of the given ideal (that is the primary decomposition of the radical of the ideal). In practice, this provides an output with much smaller coefficients, especially in the case of systems with high multiplicities.

Contrarily to triangular decompositions and equiprojectable decompositions, the RUR is not defined in positive dimension.

Algorithms for numerically solving
General solving algorithms

The general numerical algorithms which are designed for any system of simultaneous equations work also for polynomial systems. However the specific methods will generally be preferred, as the general methods generally do not allow to find all solutions. Especially, when a general method does not find any solution, this is usually not an indication that there is no solution.

Nevertheless two methods deserve to be mentioned here.

Newton's method may be used if the number of equations is equal to the number of variables. It does not allow to find all the solutions nor to prove that there is no solution. But it is very fast when starting from a point which is close to a solution. Therefore it is a basic tool for Homotopy Continuation method described below.

Optimization is rarely used for solving polynomial systems, but it succeeded, around 1970, to show that a system of 81 quadratic equations in 56 variables is not inconsistent.[10] With the other known methods this system remains beyond the possibilities of modern technology. This method consists simply in minimizing the sum of the squares of the equations. If zero is found as a local minimum, then it is attained at a solution. This method works for overdetermined systems, but outputs an empty information if all local minimums which are found are positive.

Homotopy continuation method

This is a semi-numeric method which supposes that the number of equations is equal to the number of variables. This method is relatively old but it has been dramatically improved in the last decades by J. Verschelde and his associates.[11]

This method divides into three steps. First an upper bound on the number of solutions is computed. This bound has to be as sharp as possible. Therefore it is computed by, at least, four different methods and the best value, say N, is kept.

In the second step, a system \( g_1=0, \ldots, g_n=0 \) of polynomial equations is generated which has exactly N solutions that are easy to compute. This new system has the same number n of variables and the same number n of equations and the same general structure as the system to solve, \( f_1=0, \ldots, f_n=0. \)

Then a homotopy between the two systems is considered. It consists, for example, of the straight line between the two systems, but other paths may be considered, in particular to avoid some singularities, in the system

\( (1-t)g_1+t f_1=0, \ldots, (1-t)g_n+t f_n=0. \)

The homotopy continuation consists in deforming the parameter t from 0 to 1 and following the N solutions during this deformation. This gives the desired solutions for t = 1. Following means that, if \( t_1<t_2 \), the solutions for \( t=t_2 \) are deduced from the solutions for t=t_1 by Newton's method. The difficulty here is to well choose the value of \( t_2-t_1 \): Too large, Newton's convergence may be slow and may even jump from a solution path to another one. Too small, and the number of steps slows down the method.

Numerically solving from the Rational Univariate Representation

To deduce the numeric values of the solutions from a RUR seems easy: it suffices to compute the roots of the univariate polynomial and to substitute them in the other equations. This is not so easy because the evaluation of a polynomial at the roots of another polynomial is highly unstable.

The roots of the univariate polynomial have thus to be computed at a high precision which may not be defined once for all. There are two algorithms which fulfill this requirement.

Aberth method, implemented in MPSolve computes all the complex roots to any precision.
Uspensky's algorithm of Collins and Akritas,[12] improved by Rouillier and Zimmermann [13] and based on Descartes' rule of signs. This algorithms computes the real roots, isolated in intervals of arbitrary small width. It is implemented in Maple (functions fsolve and RootFinding[Isolate]).

Software packages

There are at least four software packages which can solve zero-dimensional systems automatically (by automatically, one means that no human intervention is needed between input and output, and thus that no knowledge of the method by the user is needed). There are also several other software packages which may be useful for solving zero-dimensional systems. Some of them are listed after the automatic solvers.

The Maple function RootFinding[Isolate] takes as input any polynomial system over the rational numbers (if some coefficients are floating point numbers, they are converted to rational numbers) and outputs the real solutions represented either (optionally) as intervals of rational numbers or as floating point approximations of arbitrary precision. If the system is not zero dimensional, this is signaled as an error.

Internally, this solver, designed by F. Rouillier computes first a Gröbner basis and then a Rational Univariate Representation from which the required approximation of the solutions are deduced. It works routinely for systems having up to a few hundred complex solutions.

The rational univariate representation may be computed with Maple function

Groebner[RationalUnivariateRepresentation].

To extract all the complex solutions from a rational univariate representation, one may use MPSolve, which computes the complex roots of univariate polynomials to any precision. It is recommended to run MPSolve several times, doubling the precision each time, until solutions remain stable, as the substitution of the roots in the equations of the input variables can be highly unstable.

The second solver is PHCpack,[11][14] written under the direction of J. Verschelde. PHCpack implements the homotopy continuation method. This solver computes the isolated complex solutions of polynomial systems having as many equations as variables.

The third solver is Bertini,[15][16] written by D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler. Bertini uses numerical homotopy continuation with adaptive precision. In addition to computing zero-dimensional solution sets, both PHCpack and Bertini are capable of working with positive dimensional solution sets.

The fourth solver is the Maple command RegularChains[RealTriangularize]. For any zero-dimensional input system with rational number coefficients it returns those solutions whose coordinates are real algebraic numbers. Each of these real numbers is encoded by an isolation interval and a defining polynomial.

The command RegularChains[RealTriangularize] is part of the Maple library RegularChains, written by Marc Moreno-Maza, his students and post-doctoral fellows (listed in chronological order of graduation) Francois Lemaire, Yuzhen Xie, Xin Li, Xiao Rong, Liyun Li, Wei Pan and Changbo Chen. Other contributors are Eric Schost, Bican Xia and Wenyuan Wu. This library provides a large set of functionalities for solving zero-dimensional and positive dimensional systems. In both cases, for input systems with rational number coefficients, routines for isolating the real solutions are available. For arbitrary input system of polynomial equations and inequations (with rational number coefficients or with coefficients in a prime field) one can use the command RegularChains[Triangularize] for computing the solutions whose coordinates are in the algebraic closure of the coefficient field. The underlying algorithms are based on the notion of a regular chain.

While the command RegularChains[RealTriangularize] is currently limited to zero-dimensional systems, a future release will be able to process any system of polynomial equations, inequations and inequalities. The corresponding new algorithm[17] is based on the concept of a regular semi-algebraic system.
See also

Triangular decomposition
Wu's method of characteristic set

References

Songxin Liang, J. Gerhard, D.J. Jeffrey, G. Moroz, A Package for Solving Parametric Polynomial Systems. Communications in Computer Algebra (2009)
P. Aubry, M. Moreno Maza, Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods. J. Symb. Comput. 28, 1999
Faugère, J.C., Gianni, P., Lazard, D. and Mora, T., Efficient Computation of Zero-Dimensional Gröbner Basis by Change of Ordering. Journal of Symbolic Computation, 16, 1993
D. Lazard, Solving zero-dimensional algebraic systems. Journal of Symbolic Computation 13, 1992
Xavier Dahan and Eric Schost. Sharp Estimates for Triangular Sets. In proc. ISSAC'04, pages 103--110, ACM Press, 2004
Dahan, Xavier; Moreno Maza, Marc; Schost, Eric; Wu, Wenyuan; Xie, Yuzhen (2005). "Lifting techniques for triangular decompositions". Proceedings of ISAAC 2005. ACM Press. pp. 108–105.
Changbo Chen and Marc Moreno-Maza. Algorithms for Computing Triangular Decomposition of Polynomial Systems.In proc. ISSAC'2011, pages 83-90, ACM Press, 2011 and Journal of Symbolic Computation (to appear)
Rouillier, Fabrice (1999). "Solving Zero-Dimensional Systems Through the Rational Univariate Representation". Appl. Algebra Eng. Commun. Comput. (9).
Saugata Basu; Richard Pollack, Marie-Françoise Roy (2006). Algorithms in real algebraic geometry, chapter 12.4. Springer-Verlag.
Daniel Lazard, Thirty years of Polynomial System Solving, and now? J. Symb. Comput. 44 (2009)
Verschelde, Jan (1999). "Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation". ACM Transactions on Mathematical Software 25.
George E. Collins and Alkiviadis G. Akritas, Polynomial Real Root Isolation Using Descarte's Rule of Signs. Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation
F. Rouillier and P. Zimmerman, Efficient isolation of polynomial's real roots. Journal of Computational and Applied Mathematics 162 (2004)
Release 2.3.86 of PHCpack
Bates, D. J., Hauenstein, J. D., Sommese, A. J., & Wampler, C. W. (2013). Numerically solving polynomial systems with Bertini (Vol. 25). SIAM.
Bertini: Software for Numerical Algebraic Geometry

Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong Xiao. Triangular decomposition of semi-algebraic systems. Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187--194, 2010.

Cox, David; John Little; Donal O'Shea (1997). Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra (2nd ed.). New York: Springer. ISBN 978-0387946801.
Sturmfels, Bernd (2002). Solving systems of polynomial equations. Providence, RI: American Mathematical Soc. ISBN 0821832514.

Every Eisenstein integer a + bω whose norm a2 − ab + b2 is a rational prime is an Eisenstein prime. In fact, every Eisenstein prime is of this form, or is a product of a unit and a rational prime congruent to 2 mod 3.

This is a discrete Fourier transform.

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