    Asymptotic Solutions of Polynomial Equations with Exp-Log Coefficients

We present an algorithm for computing asymptotic approximations of roots of polynomials with exp-log function coefficients. The real and imaginary parts of the approximations are given as explicit exp-log expressions. We provide a method for deciding which approximations correspond to real roots. We report on implementation of the algorithm and present empirical data.

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1. Introduction

Definition 1.

The set of exp-log functions is the smallest set of partial functions containing , , the identity function and the constant functions, closed under addition, multiplication and composition of functions.

The domain of an exp-log function is determined as follows:

1. the domain of , the identity function and the constant functions is and the domain of is ,

2. ,

3. .

In particular, is an open set and is in .

Remark 2.

The multiplicative inverse function

 R∖{0}∋x→1/x=xexp(−log(x2))∈R

and the real exponent power functions

 R+∋x→xr=exp(rlog(x))∈R

for , are exp-log functions.

The domain of an exp-log function consists of a finite number of open, possibly unbounded, intervals and an exp-log function has a finite number of real roots. An algorithm computing domains and isolating intervals for real roots of exp-log functions is given in [11, 12].

We say that a partial function is defined near infinity if for some .

Definition 3.

A Hardy field  is a set of germs at infinity of real-valued functions that is closed under differentiation and forms a field under addition and multiplication.

Theorem 4.

[4, 3] The germs at infinity of exp-log functions defined near infinity form a Hardy field.

Let where, for , and and are exp-log functions defined near infinity ( denotes the imaginary unit).

Problem 5.

Describe the asymptotic behaviour of roots of in as tends to infinity.

The following theorem [8, 9] shows that the problem is well posed, that is the roots of in are functions in defined near infinity.

Theorem 6.

If is a Hardy field, then there exists a Hardy field such that is algebraically closed.

Corollary 7.

There exists a Hardy field such that has roots in (counted with multiplicities) in .

Definition 8.

Let be a partial function defined near infinity. We say that partial functions defined near infinity form an -term asymptotic approximation of if, for , and .

In this paper we present an algorithm which computes asymptotic approximations of roots of in . The approximations are given as exp-log expressions. The algorithm makes use of the theory of “most rapidly varying” subexpressions developed in  to compute limits of exp-log functions. In fact our algorithm applies in the more general case of MrvH fields. The algorithm is based on a Newton polygon technique [6, 14, 13] extended to “series” with arbitrary real exponents.

Algorithms given in [10, 13] solve the problem of finding asymptotic solutions of polynomial equations in more general settings. We chose to extend the algorithm of  because we find it simpler to implement and we can give a direct and elementary proof that the computed expressions satisfy our (weaker) requirements.

Example 9.

Let . One-term asymptotic approximations of roots of in computed with our algorithm are , , , ,

. Let us estimate the relative error

of the approximations, where is the exact root closest to . Using the bound

 |y−y∗|≤|5P(x,y)∂P∂y(x,y)|

on the distance from to the closest root of , after simplifications valid for , we get , and for , . Both bounds tend to zero as tends to infinity and are decreasing for , where is the principal branch of the Lambert W function. Evaluating the bounds at we get and for . For we get and for . This shows that we can obtain approximations of roots of to digits of precision by evaluating the asymptotic approximations. The evaluation takes ms. For comparison, direct computation of roots of to digits of precision takes ms. Figure 1.1 shows the asymptotic approximations of the real roots of (dashed curves) and the exact roots (solid green curves).

2. Most rapidly varying subexpressions

In this section we give a very brief summary of terminology and facts necessary for formulating Algorithm 14. We will use the algorithm to compute approximations of coefficients of in terms of a “most rapidly varying” subexpression present in the coefficients. The algorithm is based on the algorithm MrvLimit described in . For a more detailed introduction and proofs of the stated facts see .

Definition 10.

The set of exp-log expressions with coefficients in a computable field is defined recursively as follows:

1. elements of and the variable are exp-log expressions,

2. if and are exp-log expressions, so are , and ,

3. if is an exp-log expression and , then , , and are exp-log expressions.

Each exp-log expression represents an exp-log function, however the same function may be represented by many different expressions. In the following, when we refer to the domain, point values, and limits of an exp-log expression, we mean the domain, point values, and limits of the corresponding exp-log function.

Definition 11.

Let be the set of exp-log expressions such that, for some , and either as an expression or is nonzero on .

Remark 12.

Note that we exclude from expressions that are identically zero in a neighbourhood of infinity, but are not explicitly zero e.g. . The algorithm ExpLogRootIsolation of  can be used to check whether a given exp-log expression is defined near infinity and to detect expressions that are identically zero in a neighbourhood of infinity and replace them with explicit zeros. ExpLogRootIsolation requires a zero test algorithm for elementary constants. Termination of the currently known zero test algorithm relies on Schanuel’s conjecture [7, 12].

The germs at infinity of functions represented by elements of form the Hardy field of exp-log functions defined near infinity.

Theorem 13.

If and are nonzero elements of a Hardy field, then the limit

 limx→∞log|f(x)|log|g(x)|

exists (in ). Moreover, if and

 limx→∞log|f(x)|log|g(x)|=0

then, for any ,

 limx→∞f(x)g(x)e=0

The theorem follows from the results in section 3.1.2 of .

Following , we say that is more rapidly varying than , or is in a higher comparability class than , if

 limx→∞log|f(x)|log|g(x)|=0

and we denote it . We say that and have the same order of variation, or and are in the same comparability class, if

 limx→∞log|f(x)|log|g(x)|∈R∖{0}

and we denote it . We will also use to denote . is a most rapidly varying subexpression of if is a subexpression of and no subexpression of is more rapidly varying than . Let be the set of most rapidly varying subexpressions of . We will write (resp. ) if for all , (resp. ). Let be the set of such that is a subexpression of some and no subexpression of any is more rapidly varying than , that is the of (as in Algorithm 3.12 of ).

To prove termination of our algorithm we use the notion of size of an exp-log expression defined in , section 3.4.1. For an exp-log expression , let be the set of subexpressions of defined by the following conditions.

1. If does not contain the variable , then .

2. If , then .

3. If , , or , then .

4. If then .

5. If or then .

Then is defined as the cardinality of .

Let (resp. ) denote times iterated exponential (resp. logarithm), and for let (resp. ) denote with replaced with (resp. ). The following algorithm computes approximations of elements of a finite subset of in terms of their most rapidly varying subexpression.

Algorithm 14.

(MrvApprox)
Input: such that .
Output: , , , , and such that

• and ,

• ,

• if then and ,

• if then and ,

• .

The algorithm proceeds in a very similar manner to the algorithm MrvLimit described in . First, it finds the set . If the algorithm replaces with in and recomputes until . is the number of replacements performed in this step. Then the algorithm picks such that or belongs to , , and , and rewrites all elements of in terms of . If contains a subexpression in the same comparability class as , let be the first term of (as in section 3.3.3 of ), and let be the difference between the exponents of in the second and in the first term of the series ( if ). If does not contain subexpressions in the same comparability class as , then , , and . In both cases . Pick . Then is either or a power series in with positive exponents and coefficients in a lower comparability class than , hence . Section 3.4.1 of  proves that , with the strict inequality if contains a subexpression in the same comparability class as . This shows that the last requirement is satisfied.

3. Root continuity

To prove correctness of our algorithm we need a polynomial root continuity lemma that does not assume fixed degree of the polynomial. The lemma is very similar to Theorem 1 of , except that our version provides explicit bounds.

Lemma 15.

Let

 p=anzn+…+a0=an(z−r1)⋅…⋅(z−rn)∈C[z]

where and . Let and ( if ).

Suppose that , , and .

Then for every

 q=bmzm+…+b0∈C[z]

such that , for , , and, for , , we have

 q=bm(z−s1)⋅…⋅(z−sm)

for , , and for , .

Proof.

Let , , and, for , let , and . Then, for , and are either identical or disjoint, is contained in the interior of , contains exactly one of the distinct roots of , and contains all roots of . If for some , then and

 |q(c)−p(c)|≤n∑k=0|bk−ak||z|k+m∑k=n+1|bk||z|k≤δm∑k=01ϵk

We have

 δm∑k=01ϵk<|an|(1−ϵ)ϵm+n1−ϵm+11−1/ϵm+11−1/ϵ=|an|ϵn

Hence . By Rouche’s theorem, for , the number of roots of in equals the number of roots of in , which concludes the proof. ∎

4. The main algorithm

Let . This section presents the main algorithm computing asymptotic approximations of roots of polynomials .

Let us first describe a straightforward generalization of Algorithm 14 to inputs in . We extend and to by defining and . We say that (resp. ) if (resp. ). If then and .

Algorithm 16.

(MrvApproxC)
Input: such that .
Output: , , , , and such that

• and ,

• ,

• if then and ,

• if then and ,

• .

1. Let . Call Algorithm 14 with as input, obtaining , , , , and .

2. For , put and

 bi:=⎧⎪⎨⎪⎩^uieu,iev,i
3. Pick such that and for all such that .

4. Return , , , , and .

Proof.

To prove that the output of Algorithm 16 satisfies the required conditions we need to prove that if then

 limx→∞ω−(ei+d)(a↑ki−biωei)=0

The other conditions follow directly from the definitions and the properties of the output of Algorithm 14.

Suppose that . Then and

 ω−(ei+d)(a↑ki−biωei)=ω−(eu,i+d)(u↑ki−^uiωeu,i)+ıω−(eu,i+d)v↑ki

We have

 limx→∞ω−(eu,i+d)(u↑ki−^uiωeu,i)= limx→∞ω^d−dω−(eu,i+^d)(u↑ki−^uiωeu,i)=0

and

 limx→∞ω−(eu,i+d)v↑ki= limx→∞ω(ev,i−eu,i)+(^d−d)ω−(ev,i+^d)(v↑ki−^viωev,i)+ ^viω(ev,i−eu,i)−d=0

since , , and . Cases and can be proven in a similar manner. ∎

Let . W.l.o.g. we may assume that and are not identically zero.

Suppose that i.e. depends on . Let , , , , and be the output of Algorithm 16 for , and let .

Let be a Hardy field containing germs at infinity of exp-log functions defined near infinity, such that is algebraically closed. Let be a root of . Since , the limit exists.

.

Proof.

Suppose that . Then

 (4.1) 0=limx→∞Q(x,α)a↑knαn=limx→∞1+n∑i=1a↑kn−ia↑knαi

Since , , and so either and

 limx→∞|n∑i=1a↑kn−ia↑knαi|=0

or and

 limx→∞|n∑i=1a↑kn−ia↑knαi|=∞

Both cases contradict equation (4.1). ∎

Let . If , put

 ci=ω−(ei+d)(a↑ki−biωei)

Then and . Put . Then

 γ=limx→∞log|βωγ|log|ω|=γ+limx→∞log|β|log|ω|

and hence . We have

 Q(x,α)=∑i∈I(biβiωei+γi+ciβiωei+d+γi)

Let , let , and let

 μ<ν

Then

 0=ω−νQ(x,α)=ωμ−ν∑i∈Jbiβi+∑i∈I∖Jbiβiωei+γi−ν+∑i∈Iciβiωei+d+γi−ν

As tends to infinity, all terms in the last two sums tend to zero, hence

 limx→∞ωμ−ν∑i∈Jbiβi=0

Since and , the cardinality of must be at least . Consider the subset of with coordinates denoted . Then the line passes through the points and all the other points of lie above this line. This means that is the slope of one of the segments that form the lower part of the boundary of the convex hull of .

Let and . We have

 Qγ(x,z)=Rγ(x,z)+ων−μ(∑i∈I∖Jbiωei+γi−νzi+∑i∈Iciωei+d+γi−νzi)

Let be the nonzero roots of listed with multiplicities.

Claim 18.

There exist roots of such that, for sufficiently large , for , , where .

Proof.

Let be the maximum of absolute values of roots of and let be the minimum distance between two distinct roots of ( if all roots of are equal). Put and . For sufficiently large , has a fixed number of distinct roots in , equal to its number of distinct roots in . can be bounded from above by a rational function in absolute values of , for , and, since the coefficients in of are rational functions of , can be bounded from below by an expression constructed from using rational operations, square roots and absolute value (see e.g. , Theorem 5). Since , for sufficiently large , . Let be the degree of in . For sufficiently large ,

 |bs|(1−ϵ)ϵn+s1−ϵn+1>|bs|ϵn+s+1>ϵ2n+2=δ

The coefficients at , for , of have the form and , hence, for sufficiently large , the absolute value of each of these coefficients is less than . By Lemma 15, there exist roots of such that, for sufficiently large , for , . ∎

Fix and let . Then . Suppose that form an -term asymptotic approximation of such that, for , . For , put . We have

 αj−∑mi=1gigm=βj−∑mi=1fifm=βj−ρjfm+ρj−∑mi=1fifm

For sufficiently large , . Since , . Hence, , and so form an -term asymptotic approximation of . Since for any

 limx→∞f(x)=limx→∞f(logk(x))

form an -term asymptotic approximation of the root of .

Let us now consider the case where we have found an exact solution of . To simplify the description of the case let us make the following rather technical definition.

Definition 19.

Let , , , , and be the result of applying Algorithm 16 to . We will call a root of asymptotically small if and (in other words, with ).

Suppose that we have found an exact solution of of multiplicity . Then there exist exactly roots of such that, for sufficiently large , for , . Hence, there exist exactly roots of such that, for sufficiently large , for , . The mapping is a bijection between the roots of such that, for sufficiently large , and the roots of such that . Since can be chosen arbitrarily close to , can be arbitrarily small. Therefore the mapping