Complex Polynomial Systems

Introduction

Mathematica functions Reduce, Resolve, and FindInstance allow to solve a wide variety of problems that can be expressed in terms of equations and inequalities. The functions use a collection of algorithms applicable to classes of problems satisfying particular properties, as well as a set of heuristics that attempt to reduce the given problem to a sequence of problems which can be solved using the algorithms. This notebook describes the algorithms used to solve the class of problems known as complex polynomial systems. It characterizes the structure of the returned answers, and describes the options that affect various aspects of solving the complex polynomial systems.

A complex polynomial system is an expression constructed with polynomial equations and inequations

combined using logical connectives and quantifiers

An occurrence of a variable x inside is called a bound occurrence, and any other occurrence of x is called a free occurrence. A variable x is called a free variable of a complex polynomial system if the system contains a free occurrence of x. A complex polynomial system is quantifier-free if it contains no quantifiers.

Here is an example of a complex polynomial system with free variables x, y, and z.

In Mathematica, quantifiers are represented using the functions Exists () and ForAll ().

Any complex polynomial system can be transformed to the prenex normal form

where each is a quantifier or , and is quantifier-free.

Any quantifier-free complex polynomial system can be transformed to the disjunctive normal form

where each is a polynomial equation or inequation.

Reduce, Resolve, and FindInstance always put complex polynomial systems in the prenex normal form, with quantifier-free parts in the disjunctive normal form, and subtract the sides of the equations and inequations to put them in the form

In the following sections, assume that the system has been transformed to this form.

Reduce can solve arbitrary complex polynomial systems. The solution (possibly after expanding with respect to ) is a disjunction of terms of the form

where are the free variables of the system, each is a polynomial, each is an algebraic function expressed using radicals or Root objects, and any terms of the conjunction (4) may be absent. Each is well defined, that is, no denominators or leading terms of Root objects in become zero for any satisfying the preceding terms of the conjunction (4).

This solves the system (1).

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Resolve can eliminate quantifiers from arbitrary complex polynomial systems. If no variables are specified, the result is a logical combination of terms

where f and g are polynomials, and each is a free variable of the system. With variables specified in the input, Resolve gives the same answer as Reduce.

This eliminates quantifiers from the system (1).

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FindInstance can handle arbitrary complex polynomial systems giving instances of complex solutions, or an empty list for systems that have no solutions. If the number of instances requested is more than one, the instances are randomly generated from the full solution of the system, and therefore they may depend on the value of the RandomSeed option. If one instance is requested, a faster algorithm that produces one instance is used, and the instance returned is always the same.

This finds a solution for the system (1).

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The main tool used in solving complex polynomial systems is the Gröbner basis algorithm (see, e.g., [1]), which is directly available in Mathematica as the GroebnerBasis function.

Gröbner Bases

Theory

This section gives a very brief introduction to the theory of Gröbner bases. It presents only the properties that are necessary to describe the algorithms used by Mathematica in solving complex polynomial systems. For a more complete presentation see, for example, [1], [2]. Note that what [2] calls a monomial, [1] calls a term, and vice versa. This notebook uses the terminology of [1].

A monomial in is an expression of the form with nonnegative integers .

Let be the set of all monomials in . A monomial order is a linear order on M, such that 1 t for all t M, and implies for all .

Let R be a field, the domain of integers, or the domain of univariate polynomials over a field. Let Quot and Rem be functions defined as follows. If R is a field, Quot(a, b) = a/b, and Rem(a, b) = 0. If R is the domain of integers, Quot and Rem are the integer quotient and remainder functions, with . If R is the domain of univariate polynomials over a field, Quot and Rem are the polynomial quotient and remainder functions.

A product , where a is a nonzero element of R and m is a monomial, is called a term.

Let be a monomial order on M, and let . The leading monomial LM(f) of f is the -largest monomial appearing in f, the leading coefficient LC(f) of f is the coefficient at LM(f) in f, and the leading term LT(f) of f is the product LC(f) LM(f).

A Gröbner basis of an ideal I in , with respect to a monomial order , is a finite set G of polynomials, such that for each f I, there exists g G, such that LT(g) divides LT(f). Every ideal I has a Gröbner basis (see, e.g., [1] for a proof).

Let , and let be a monomial. A term is reducible modulo p, if LM(p) divides m, and a ≠ Rem(a, LC(p)). If t is reducible modulo p, the reduction of t modulo p is the polynomial

Note that if Rem(a, LC(p)) ≠ 0, then LT(Red(t, p)) = Rem(a, LC(p)) m; otherwise LM(Red(t, p)) m.

Let , and let P be an ordered finite subset of . f is reducible modulo P if f contains a term reducible modulo an element of P. The reduction Red(f, P) of f modulo P is defined by the following procedure. While the set RT of terms of f reducible modulo an element of P is not empty, take the term t RT with the -largest monomial, take the first p P, such that t is reducible modulo p, and replace the term t in f with Red(t, p). Note that the monomials of terms t chosen in subsequent steps of the procedure form a -descending chain, and each monomial can appear at most k times, where k is the number of elements of P, hence the procedure terminates.

A Gröbner basis G is semi-reduced if for all g G, g is not reducible modulo G\{g}, and if R is the domain of integers, .

The Mathematica function GroebnerBasis returns semi-reduced Gröbner bases. In the following discussion, all Gröbner bases are assumed to be semi-reduced. Note that this is not the same as reduced Gröbner bases defined in the literature, since here the basis polynomials are not required to be monic. For a fixed monomial order, every ideal has a unique reduced Gröbner basis. Semi-reduced Gröbner bases defined here are only unique up to multiplication by invertible elements of R (see Property 2).

Property 1: Let G be a Gröbner basis of an ideal I in , and let . Then iff Red(f, G) = 0.

This is a simple consequence of the definitions.

Property 2: Let and be two Gröbner bases of an ideal I with respect to the same monomial order , and suppose that elements of G and H are ordered by their leading monomials. Then k = m, and for all , if R is the domain of integers, , otherwise for some invertible element of R.

If LM(f) = LM(g), then LT(f) is reducible modulo g or LT(g) is reducible modulo f. Hence the leading monomials of the elements of a Gröbner basis are all different. Without loss of generality assume k ≤ m. For induction, fix j ≤ k and suppose that for all i < j, for some invertible element of R. If R is the domain of integers, . Without loss of generality assume . Since belongs to I, there exists i, such that divides . Then , and so . If , then would be reducible modulo and also modulo , which is impossible, since G is semi-reduced. Hence , and , and divides . Similarly, divides . Therefore, there exists an invertible element of R, such that . If R is the domain of integers, and are positive, and so . Let . Suppose . Since r belongs to I, LT(r) must be divisible by , for some i < j. Let and be the coefficients at LM(r) in and . If R is a field, the term of is reducible modulo , which contradicts the assumption that G is semi-reduced. If R is the domain of univariate polynomials over a field,

and so either is reducible modulo , or is reducible modulo , which contradicts the assumption that G and H are semi-reduced. Finally, let R be the domain of integers. Since neither is reducible modulo nor is reducible modulo , and . Hence , which is impossible, since LT(r) is divisible by . Therefore , and so . By induction on j, for all j ≤ k, . If , then would be reducible modulo some , with j ≤ k, and hence would be reducible modulo . Therefore , which completes the proof of Property 2.

Property 3: Let I be an ideal in , let , and let G be a Gröbner basis of the ideal in . Then f belongs to the radical of I iff for an invertible element c of R.

If an ideal contains invertible elements of R, GroebnerBasis always returns {1}.

Note first that

belongs to the ideal for any nonnegative integer k. Hence, if f belongs to the radical of I, then 1 belongs to J. Since G is a Gröbner basis of J, it must contain an element c whose leading coefficient divides 1. Hence c is an invertible element of R. Since G is semi-reduced and c divides any term, . Now suppose that for an invertible element c of R. Then 1 belongs to J, and so

where each belongs to I, and each belongs to . Hence comparing coefficients at powers of y leads to the following equations modulo I: , , for , and . Then, , for , and modulo I. Therefore, f belongs to the radical of I, which completes the proof of Property 3.

The following more technical property is important for solving complex polynomial systems.

Property 4: Let G be a Gröbner basis of an ideal I in with a monomial order that makes monomials containing y greater than monomials not containing y, let h be the element of G with the lowest positive degree d in y, let be the leading coefficient of h in y, and let be all elements of G that do not depend on y. Then for any polynomial p I and any point if , , for , and , then .

Consider the pseudoremainder r of the division of by h as polynomials in y.

Since p and h belong to I, so does r. By Property 1, reduction of r by G must yield zero. Since the degree of r in y is less than d, r cannot be reduced by any of the elements of G that depend on y. Hence

and so . Since , (5) implies that , which completes the proof of Property 4.

Mathematica function GroebnerBasis

The Mathematica function GroebnerBasis finds semi-reduced Gröbner bases. This section describes GroebnerBasis options used in solving of complex polynomial systems.

CoefficientDomain

This option specifies the domain R of coefficients. With the default Automatic setting, the coefficient domain is the field generated by numeric coefficients present in the input.

Note that the coefficient domain R also depends on the setting of the Modulus option of GroebnerBasis. With Modulus->p, for a prime number p, the coefficient domain is the field , or the field of rational functions over if CoefficientDomain->RationalFunctions.

Method

With the default setting Method->Buchberger, GroebnerBasis uses a variant of the Buchberger algorithm. Another algorithm available is the Gröbner walk, which computes a Gröbner basis in an easier monomial order and then transforms it to the required harder monomial order. This is often faster than directly computing a Gröbner basis in the required order, especially if the input polynomials are known to be a Gröbner basis for the easier order.

MonomialOrder

This option specifies the monomial order. The value can be either one of the named monomial orders or a weight matrix. The following table gives conditions for .

Quantifier elimination needs an order in which monomials containing quantifier variables are greater than monomials not containing quantifier variables. The Lexicographic order satisfies this condition, but the following EliminationOrder usually leads to faster computations.

X denotes free variables, Y denotes quantifier variables, and are monomials, and denotes the DegreeReverseLexicographic order.

Using EliminationOrder requires the GroebnerBasis syntax with elimination variables specified.

By default, GroebnerBasis with MonomialOrder->EliminationOrder, drops the polynomials that contain yvars from the result, returning only basis polynomials in xvars. To get all basis polynomials, the value of the system option EliminateFromGroebnerBasis from the GroebnerBasisOptions group must be changed. (Mathematica changes the option locally in the quantifier elimination algorithm.) The option value can be changed with

This eliminates y from . The answer is a polynomial whose zeros are the Zariski closure of the projection of the solution set of the two original equations on the plane.

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The exact description of the projection of the solution set on the plane depends on all basis polynomials. Note that the second basis polynomial cannot be zero if or are zero.

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This resets the system option to its default value.

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Resolve gives the exact description of the projection of the solution set on the plane.

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Decision Problems

A decision problem is a system with all variables existentially quantified, that is, a system of the form

where are all variables in . Solving a decision problem means deciding whether it is equivalent to True or to False, that is, deciding whether the quantifier-free system of polynomial equations and inequations has solutions.

Solving this decision problem proves that a quadratic equation with a zero determinant cannot have two different roots.

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Given the identities

solving any decision problem can be reduced to solving a finite number of decision problems of the form

By Hilbert's nullstellensatz and Property 3 of Gröbner bases

has complex solutions iff

with an arbitrary monomial order, is different than {1}.

This shows that has complex solutions.

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This shows that has no complex solutions.

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When Mathematica solves a decision problem, the monomial order used by the GroebnerBasis computation is MonomialOrder->EliminationOrder, with {z} specified as the elimination variable list. This setting corresponds to the monomial ordering in which monomials containing z are greater than those that do not contain z, and the ordering of monomials not containing z is degree reverse lexicographic. If there is no inequation condition, there is no need to introduce z, and Mathematica uses MonomialOrder->DegreeReverseLexicographic.

Quantifier Elimination

For any complex polynomial system there exists an equivalent quantifier-free complex polynomial system. This follows from Chevalley's theorem stating that a projection of a quasi-algebraically constructible set (a solution set of a quantifier-free system of polynomial equations and inequations) is a quasi-algebraically constructible set (see [3]). Quantifier elimination is the procedure of finding a quantifier-free complex polynomial system equivalent to a given complex polynomial system. In Mathematica, quantifier elimination for complex polynomial systems is done by Resolve. It is also used by Reduce and FindInstance as the first step in solving or finding instances of solutions of complex polynomial systems.

Eliminating quantifiers from this system gives a condition for quadratic equations to have at least two different zeros.

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For complex polynomial systems Mathematica uses the following quantifier elimination method. Given the identities

eliminating quantifiers from any complex polynomial system can be reduced to a finite number of single existential quantifier eliminations from systems of the form

To eliminate the quantifier from (6), Mathematica first computes the Gröbner basis of equations

with a monomial order that makes monomials containing y greater than monomials not containing y.

The monomial order used is EliminationOrder, with {y} specified as the elimination variable list, and with all basis polynomials kept.

If G contains no polynomials that depend on y, then a quantifier-free system equivalent to (6) can be obtained by equating all elements of G to zero, and asserting that at least one coefficient of g as a polynomial in y is not equal to zero. Otherwise let h be the element of G with the lowest positive degree d in y, let be the leading coefficient of h in y, and let be all elements of G that do not depend on y. Now (6) can be split into a disjunction of two systems

and

To eliminate the quantifier from (7), the quantifier elimination procedure is called recursively. Since the ideal generated by strictly contains the ideal generated by , the Noetherian property of polynomial rings guarantees finiteness of the recursion.

If c belongs to the radical of the ideal generated by , which is exactly when 1 belongs to

(8) is equivalent to False. Otherwise let

be the pseudoremainder of the division of by h as polynomials in y. Then (8) is equivalent to the quantifier-free system

To show that (8) implies (9), suppose that satisfies (8). Then and there exists b, such that

Since and h belong to the ideal generated by ,

and . Hence

which implies that

To show that (9) implies (8), suppose that satisfies (9). Then

Since is a polynomial of degree d, and is a nonzero polynomial of degree less than d, there is a root b of such that divides but not for some . If was zero, then would divide , which is impossible because it would imply that divides . Therefore . Property 4 shows that for any polynomial p G. Since G is a Gröbner basis of the ideal generated by ,

which completes the proof of correctness of the quantifier elimination algorithm.

This eliminates the quantifier from . Here , , and . Since c is a nonzero constant, (7) is and the equivalent quantifier-free system is given by (9). Since g is a nonzero constant, (9) becomes .

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This resets the system option to its default value.

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Arbitrary Complex Polynomial Systems

FindInstance

FindInstance can handle arbitrary complex polynomial systems giving instances of complex solutions, or an empty list for systems that have no solutions. If the number of instances requested is more than one, the instances are randomly generated from the full solution of the system given by Reduce. If one instance is requested, a faster algorithm that produces one instance is used. Here is a description of the algorithm used to find a single instance, or prove that a system has no solutions.

If the system contains general quantifiers (), the quantifier elimination algorithm is used to eliminate the innermost quantifiers until the system contains only existential quantifiers () or is quantifier-free. Note that

has solutions if and only if has solutions, and if is a solution of then is a solution of (10). Hence to find instances of solutions of systems containing only existential quantifiers it is enough to be able to find instances of quantifier-free systems. Moreover, is a solution of

if and only if it is a solution of one of the , with , so it is enough to show how to find instances of solutions of

First compute the GroebnerBasis G of with MonomialOrder->EliminationOrder, eliminating the polynomials that depend on z (if there is no inequation condition, G is the GroebnerBasis of with MonomialOrder->DegreeReverseLexicographic). If G contains 1, there are no solutions. Otherwise, compute a subset S of of the highest cardinality among subsets strongly independent modulo the ideal generated by G with respect to the degree reverse lexicographic order (see [1], Section 9.3). Reorder so that , and compute the lexicographic order GroebnerBasis of the ideal generated by G. To compute , Mathematica uses the Gröbner walk algorithm.

For each of the variables , , select the polynomial with the smallest leading monomial among elements of that depend on and not on . Let be the leading coefficient of as a polynomial in . If depends on a variable that is not in S, replace with the lexicographic order Gröbner basis of the ideal generated by and . The following shows that this operation keeps S strongly independent modulo the ideal generated by . Hence, possibly after a finite (by the Noetherian property of polynomial rings) number of extensions of , the leading coefficient of depends only on , for all . For the set of polynomials P, let Z(P) be the set of common zeros of elements of P. Both Z(G) and Z(H) have dimension d, and , hence any d-dimensional irreducible component of Z(H) is also a component of Z(G). Since g does not vanish on any irreducible component of Z(G), it does not vanish on any d-dimensional irreducible component of Z(H). Therefore, the Gröbner basis of H and g contains a polynomial t depending only on . Let . To find a solution of (11), pick its last d coordinates so that . For all , , and so by Property 4 if , for , is chosen to be the first root of , then . Moreover, , because otherwise would belong to , which would imply that , which is impossible since t divides p.

To prove the correctness of the aforementioned algorithm, it must be shown that extending H by that depend on a variable not in S preserves strong independence of S modulo the ideal generated by . Suppose for some , depends on a variable, which is not in S. Let