Heine–Stieltjes polynomials

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 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

$\{\begin{array}{ll}{s}^3+4c^3-3c& =0\\ s^2+c^2-1& =0\end{array}$

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 but not inconsistent.

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. 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 side 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 a lot of 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 in 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

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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

$\{\begin{array}{l}{f}_1(x_1)=0\\ f_2(x_1,x_2)=0\\ \cdots \\ f_n(x_1,x_2,\dots ,x_n)=0\end{array}$

The solutions of this system are obtained by solving the first univariate equations, substitute 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 this 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.

$\{\begin{array}{l}{x}^2-1=0\\ (x-1)(y-1)=0\\ y^2-1=0\end{array}$

There are several algorithms for computing a triangular decomposition of an arbitrary polynomial system (not necessarily zero-dimensional) 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 and finally applying the Lextriangular algorithm.

This representation of the solutions and the algorithms to compute it are presently, in practice, a very efficient way for solving zero-dimensional polynomial systems with coefficients in a finite field.

For rational coefficients, the Lextriangular algorithm has two drawbacks:

• 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.

Most algorithms computing triangular decompositions directly (that is, without precomputing a Gröbner Basis) share above drawbacks, but the most recent ones do not suffer from the one related to output size, as shown by the experimental results reported by Changbo Chen and M. Moreno-Maza. Actually, this observation is predicted by a theoretical argument (which does not give rise to a practical algorithm, though): For a given polynomial system $S$ whose solutions can be described by a single regular chain, there exists one regular chain representing $S$ in a nearly optimal way in term of size.

In order to address both drawbacks, one can take advantage of the rational univariate representation, which follows. Its output is a single regular chain whose coefficient size is also nearly optimal. However, if the set of solutions has several components of various multiplicities, an output of smaller size may be obtained by decomposing it first with a triangular decomposition algorithm.

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 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

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

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 g_i 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 separating variable, then the RUR is

$\{\begin{array}{l}{t}^3-t=0\\ x=\frac{t^2+2t-1}{3t^2-1}\\ y=\frac{t^2-2t-1}{3t^2-1}\\ \end{array}$

The RUR is uniquely defined for a given separating element, independently of any algorithm and it preserves the information on the multiplicities of the roots. Basically, a triangular decomposition of a zero-dimensional system does not preserve the multiplicities and is not uniquely defined, but, among all triangular decompositions of a given zero-dimensional system $S$, the equiprojectable decomposition depends only on a coordinate choice of $S$. For this latter, as for the RUR, sharp bounds are available for the coefficients. Consequently, efficient algorithms, based on so-called modular methods, exist for computing the equiprojectable decomposition and the RUR.

These bounds can trivially been obtained for complete intersection systems for the RUR by simply deriving the u-resultant associated with the system, which gives a quite direct way to bound those of an equiprojectable decomposition which are more or less equivalent.

On the computational point of view, there is one main difference between the equiprojectable decomposition and the RUR. The latter has the conceptual advantage of reducing the numeric computation of the solutions to computing the roots of a single univariate polynomial and substituting in some rational functions. One can easily show that the required computation time is then dominated by the isolation of the roots of the univariate polynomial and their refinement up to a sufficient precision.

Moreover, the RUR can trivially been decomposed to get a primary decomposition of the system and, in practice, to get much smaller coefficients than the non decomposed form, especially in the case of systems with high multiplicities. In short one can provide instantaneously a RUR of each primary component through a squarefree decomposition of the first polynomial.

On the other hand, one has to retain that triangular decomposition can be performed in positive dimension, which is not the case of the RUR.

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 not 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. 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.

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 $g_1=0,\dots ,g_n=0$ 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, $f_1=0,\dots ,f_n=0$.

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

$(1-t)g_1+t{f}_1=0,\dots ,(1-t)g_n+t{f}_n=0$.

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, improved by Rouillier and Zimmermann 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 three 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, 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 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 is based on the concept of a regular semi-algebraic system.

See also

• Triangular decomposition
• Wu's method of characteristic set

References

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