Complex Polynomial Systems

The Mathematica functions Reduce, Resolve, and FindInstance allow you 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 that can be solved using the algorithms. This tutorial 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 the methods involved.

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 _x or _x 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 Q_i is a quantifier or , and (x₁, ..., x_n;y₁, ..., y_m) is quantifier-free.

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

where each _{i, j} 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 all the tutorials for complex polynomial system solving, 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 x₁, ..., x_n are the free variables of the system, each g_i is a polynomial, each r_i is an algebraic function expressed using radicals or Root objects, and any terms of the conjunction (2) may be absent. Each r_i (x₁, ..., x_i-1) is well defined, that is, no denominators or leading terms of Root objects in r_i become zero for any (x₁, ..., x_i-1) satisfying the preceding terms of the conjunction (2).

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 x_i is a free variable of the system. With variables specified in the input, Resolve gives the same answer as Reduce.

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.

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

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 tutorial uses the terminology of [1].

A monomial in x₁, ..., x_n is an expression of the form x₁^e₁...x_n^e_n with non-negative integers e_i.

Let M=M (x₁, ..., x_n) be the set of all monomials in x₁, ..., x_n. A monomial order is a linear order on M, such that 1t for all tM, and t₁t₂ implies t₁st₂s for all t₁, t₂, s M.

Let R be a field, the domain of integers, or the domain of univariate polynomials over a field. Let Quot and Rem be functions R²R 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 -|b|/2<Rem (a, b)≤|b|/2. If R is the domain of univariate polynomials over a field, Quot and Rem are the polynomial quotient and remainder functions.

A product t=a m, 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 fR[x₁, ..., x_n]\{0}. 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 R[x₁, ..., x_n], with respect to a monomial order , is a finite set G of polynomials, such that for each fI, there exists gG, such that LT (g) divides LT (f). Every ideal I has a Gröbner basis (see [1] for a proof).

Let pR[x₁, ..., x_n]\{0}, and let mR[x₁, ..., x_n] be a monomial. A term t=a m 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 fR[x₁, ..., x_n], and let P be an ordered finite subset of R[x₁, ..., x_n]\{0}. 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 tRT with the -largest monomial, take the first pP, 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 gG, g is not reducible modulo G\{g}, and if R is the domain of integers, LC (g)>0.

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 R[x₁, ..., x_n], and let fR[x₁, ..., x_n]. Then fI iff Red (f, G)=0.

This is a simple consequence of the definitions.

Property 2: Let G={g₁, ...g_k} and H={h₁, ...h_m} 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 1≤i≤k, if R is the domain of integers, g_i=h_i; otherwise, g_i=c_i h_i for some invertible element c_i of R.

Proof: 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, g_i=c_i h_i for some invertible element c_i of R. If R is the domain of integers, c_i=1. Without loss of generality, assume LM (g_j)LM (h_j). Since g_j belongs to I, there exists i such that LT (h_i) divides LT (g_j). Then LM (h_i)LM (g_j), and so i≤j. If i<j, then g_j would be reducible modulo h_i and also modulo g_i=c_i h_i, which is impossible, since G is semi-reduced. Hence i=j, and LM (g_j)=LM (h_j), and LT (h_j) divides LT (g_j). Similarly, LT (g_j) divides LT (h_j). Therefore, there exists an invertible element c_j of R, such that LT (g_j)=c_jLT (h_j). If R is the domain of integers, LC (g_j) and LC (h_j) are positive, and so c_j=1. Let r=c_jh_j-g_j. Suppose r≠0. Since r belongs to I, LT (r) must be divisible by LT (g_i), for some i<j. Let and be the coefficients at LM (r) in g_j and h_j. If R is a field, the term LM (r) of g_j is reducible modulo g_i, which contradicts the assumption that G is semi-reduced. If R is the domain of univariate polynomials over a field,

and so either g_j is reducible modulo g_i, or h_j is reducible modulo h_i=c_ig_i, which contradicts the assumption that G and H are semi-reduced. Finally, let R be the domain of integers. Since neither g_j is reducible modulo g_i nor h_j is reducible modulo h_i=g_i, -LC (g_i)/2<≤LC (g_i)/2 and -LC (g_i)/2< ≤LC (g_i)/2. Hence -LC (g_i)<LC (r)=-<LC (g_i), which is impossible, since LT (r) is divisible by LT (g_i). Therefore r=0, and so g_j=c_j h_j. By induction on j, for all j≤k, g_j=c_j h_j. If k<m, then h_k+1 would be reducible modulo some g_j, with j≤k, and hence h_k+1 would be reducible modulo h_j=c_j^-1g_j. Therefore k=m, which completes the proof of Property 2.

Property 3: Let I be an ideal in R[x₁, ..., x_n], let fR[x₁, ..., x_n], and let G be a Gröbner basis of the ideal <I, 1-y f> in R[x₁, ..., x_n, y]. Then f belongs to the radical of I iff G={c} for an invertible element c of R.

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

Proof: Note first that

belongs to the ideal J=<I, 1-y f> for any non-negative 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, G={c}. Now suppose that G={c} for an invertible element c of R. Then 1 belongs to J, and so

where each a_i belongs to I, and each b_i belongs to R[x₁, ..., x_n]. Hence comparing coefficients at powers of y leads to the following equations modulo I: b₀1, b_ib_i-1f, for 1≤i≤m-1, and b_m-1f0. Then, b_ifⁱ, for 0≤i≤m-1, and f^m0 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 [x₁, ..., x_n, y] 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 c (x₁, ..., x_n) be the leading coefficient of h in y, and let {h₁, ..., h_s} be all elements of G that do not depend on y. Then for any polynomial pI and any point (a₁, ..., a_n, b) if c (a₁, ..., a_n)≠0, h_i (a₁, ..., a_n)=0, for 1≤i≤s, and h (a₁, ..., a_n, b)= 0, then p (a₁, ..., a_n, b)= 0.

Proof: Consider the pseudoremainder r of the division of p 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 r (a₁, ..., a_n, b)=0. Since c (a₁, ..., a_n)≠0, (1) implies that p (a₁, ..., a_n, b)= 0, which completes the proof of Property 4.

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

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 _p, or the field of rational functions over _p if CoefficientDomain->RationalFunctions.

With the default setting Method->Automatic, GroebnerBasis normally 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. With the Method->Automatic setting, GroebnerBasis uses the Gröbner walk for the default CoefficientDomain->Rationals and MonomialOrder->Lexicographic.

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 x₁^d₁...x_n^d_n x₁^e₁...x_n^e_n.

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.

where d denotes total degree, X denotes free variables, Y denotes quantifier variables, m_i and n_i are monomials, and _DRL 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

SetSystemOptions["GroebnerBasisOptions"->{"EliminateFromGroebnerBasis"->False}].

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

where x₁, ..., x_n 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 (x₁, ..., x_n) has solutions.

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}.

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.

For any complex polynomial system there exists an equivalent quantifier-free complex polynomial system. This follows from Chevalley's theorem, which states 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 [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.

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 (1), 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 all basis polynomials kept.

If G contains no polynomials that depend on y, then a quantifier-free system equivalent to (1) 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 c (x₁, ..., x_n) be the leading coefficient of h in y, and let {h₁, ..., h_s} be all elements of G that do not depend on y. Now (1) can be split into a disjunction of two systems

and

To eliminate the quantifier from (2), the quantifier elimination procedure is called recursively. Since the ideal generated by {c, f₁, ..., f_k} strictly contains the ideal generated by {f₁, ..., f_k}, the Noetherian property of polynomial rings guarantees finiteness of the recursion.

If c belongs to the radical of the ideal generated by {f₁, ..., f_k}, which is exactly when 1 belongs to

(3) is equivalent to False. Otherwise let

be the pseudoremainder of the division of g^d by h as polynomials in y. Then (3) is equivalent to the quantifier-free system

To show that (3) implies (4), suppose that (a₁, ..., a_n) satisfies (3). Then c (a₁, ..., a_n)≠ 0 and there exists b, such that

Since {h₁, ..., h_s} and h belong to the ideal generated by {f₁, ..., f_k},

and h (a₁, ..., a_n, b)=0. Hence

which implies that

To show that (4) implies (3), suppose that (a₁, ..., a_n) satisfies (4). Then

Since h (a₁, ..., a_n, y) is a polynomial of degree d, and r (a₁, ..., a_n, y) is a nonzero polynomial of degree less than d, there is a root b of h (a₁, ..., a_n, y) such that (y-b)^m divides h (a₁, ..., a_n, y) but not r (a₁, ..., a_n, y) for some 1≤m≤d. If g (a₁, ..., a_n, b) was zero, then (y-b)^m would divide g (a₁, ..., a_n, y)^d, which is impossible because it would imply that (y-b)^m divides r (a₁, ..., a_n, y). Therefore g (a₁, ..., a_n, b)≠0. Property 4 shows that p (a₁, ..., a_n, b)= 0 for any polynomial pG. Since G is a Gröbner basis of the ideal generated by {f₁, ..., f_k},

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

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 (x₁, ..., x_n, y₁, ..., y_m) has solutions, and if (a₁, ..., a_n, b₁, ..., b_m) is a solution of (x₁, ..., x_n, y₁, ..., y_m), then (b₁, ..., b_m) is a solution of (1). 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, (a₁, ..., a_n) is a solution of

if and only if it is a solution of one of the _i (x₁, ..., x_n), with 1≤i≤m, so it is enough to show how to find instances of solutions of

First compute the GroebnerBasis G of {f₁, ..., f_k, 1-g z} with MonomialOrder->EliminationOrder, eliminating the polynomials that depend on z (if there is no inequation condition, G is the GroebnerBasis of {f₁, ..., f_k} with MonomialOrder->DegreeReverseLexicographic). If G contains 1, there are no solutions. Otherwise, compute a subset S of {x₁, ..., x_n} of the highest cardinality among subsets strongly independent modulo the ideal generated by G with respect to the degree reverse lexicographic order ([1], Section 9.3). Reorder {x₁, ..., x_n} so that S={x_n-d+1, ..., x_n}, and compute the lexicographic order GroebnerBasis H of the ideal generated by G. To compute H, Mathematica uses the Gröbner walk algorithm.

For each of the variables x_i, 1≤i≤n-d, select the polynomial h_iH with the smallest leading monomial among elements of H that depend on x_i and not on {x₁, ..., x_i-1}. Let c_i be the leading coefficient of h_i as a polynomial in x_i. If c_i depends on a variable that is not in S, replace H with the lexicographic order Gröbner basis of the ideal generated by H and c_i. The following shows that this operation keeps S strongly independent modulo the ideal generated by H. Hence, possibly after a finite (by the Noetherian property of polynomial rings) number of extensions of H, the leading coefficient c_i of h_i depends only on {x_n-d+1, ..., x_n}, for all 1≤i≤n-d. 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 Z (H)Z (G), 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 {x_n-d+1, ..., x_n}. Let p=t c₁...c_n-d. To find a solution of (2), pick its last d coordinates {a_n-d+1, ..., a_n} so that p (a_n-d+1, ..., a_n)≠0. For all 1≤i≤n-d, c_i (a_n-d+1, ..., a_n)≠0, and so by Property 4 if a_i, for i=n-d, ..., 1, is chosen to be the first root of h_i (x_i, a_i+1, ..., a_n), then (a₁, ..., a_n)Z (H)Z (G). Moreover, g (a₁, ..., a_n)≠0, because otherwise (a₁, ..., a_n) would belong to Z (H{g}), which would imply that t (a_n-d+1, ..., a_n)= 0, which is impossible since t divides p.

To prove the correctness of the aforementioned algorithm, it must be shown that extending H by c_i that depend on a variable not in S preserves strong independence of S modulo the ideal generated by H. Suppose for some 1≤i≤n-d, c_i depends on a variable, which is not in S. Let I_i+1[x_i+1, ..., x_n] denote the ideal generated by H[x_i+1, ..., x_n], and let J_i+1[x_i+1, ..., x_n] denote the ideal generated by I_i+1 and c_i. Then J_i+1 does not contain nonzero elements of [x_n-d+1, ..., x_n]. To prove this, suppose that r=p c_i+qJ_i+1[x_n-d+1, ..., x_n]\{0} where p[x_i+1, ..., x_n] and qI_i+1. Then h_i= c_ix_i^k+t, with deg_xi (t)<k, and

belongs to the ideal generated by H, and so does g_i= r x_i^k+p t. This contradicts the choice of h_i since the leading monomial of g_i depends on x_i and is strictly smaller than the leading monomial of h_i. Therefore, the projection of Z (J_i+1) on A_d= (^d)_{{xn-d+1, ..., xn}} is dense in A_d, and so, since Z (I_i+1) has dimension d, c_i must be zero on some irreducible component C_i+1 of Z (I_i+1) whose projection on A_d is dense in A_d. Since Z (I_i+1) is the Zariski closure of the projection of the d-dimensional set Z (H), C_i+1 is contained in the Zariski closure of the projection of an irreducible component C of Z (H). Z (c_i)C has dimension d, hence c_i is zero on C, and the projection of C on A_d is dense in A_d, which proves that S is strongly independent modulo the ideal generated by H and c_i.

Reduce can solve arbitrary complex polynomial systems. As the first step, Reduce uses the quantifier elimination algorithm to eliminate the quantifiers. If the obtained quantifier-free system is a disjunction, each term of the disjunction is solved separately, and the solution is given as a disjunction of the solutions of the terms. Thus, the problem is reduced to solving quantifier-free systems of the form

First compute the GroebnerBasis G of {f₁, ..., f_k, 1-g z} with variable order {z, x_n, ..., x₁} and MonomialOrder->Lexicographic, and select the polynomials that do not depend on z. Then the solution set of G=0 g (x₁, ..., x_n)≠0 is equal to the solution set of (3) and g does not vanish on any component of the zero set Z (G) of G. If G contains 1, (3) has no solutions. Otherwise for each 1≤i≤n, such that the set G_i of elements of G depending on x_i and not on any x_j with j>i is not empty, select an element h_i of G_i with the lowest positive degree in x_i. If one of the leading coefficients c_i of h_i is zero on Z (G), that is, it belongs to the radical of the ideal generated by G, replace G by the lexicographic Gröbner basis of the ideal generated by G and c_i. Now split the system into

and call the solving procedure recursively on all but the last term of the disjunction (4). Note that the algebraic set c_ij=0G=0 is strictly contained in G=0, so the recursion is finite. If the product of all the c_i and g belongs to the radical of the ideal generated by G, the last term has no solutions. Otherwise, by Property 4, the solution set of the last term is equal to

The conditions c_ij≠0 guarantee that all the solutions (represented as radicals or Root objects) given by Roots[h_ij=0, x_ij] are well defined. Reduce performs several operations in order to simplify the inequation conditions returned, like removing multiple factors, removing factors common with earlier inequation conditions, reducing modulo the h_ij, and removing factors that are nonzero on Z (G).

The Mathematica functions for solving complex polynomial systems have a number of options that control the way they operate. This section gives a summary of these options.

Here are the system options from the ReduceOptions group that may affect the behavior of Reduce, Resolve, and FindInstance for complex polynomial systems. The options can be set with

SetSystemOptions["ReduceOptions"->{"option name"->value}].