Abstract
The problem of rationalizing denominators for two types of fractions is discussed in the paper. By using the theory and algorithms of Gröbner bases, we first introduce a method to rationalize the denominators of fractions with square root and cube root, and then, for the denominators with higher radical of the general form, the problem of rationalizing denominators is converted into the related problem of finding the minimal polynomials. Some interesting results and an executable algorithm for rationalizing the denominator of these type fractions are presented. Furthermore, an example is also established to illustrate the effectiveness of the algorithm.
1. Introduction
A typical topic in algebra is rationalizing denominators [1]. Rationalizing the denominator avoids the problem known as “subtrative cancellation,” deals with the problem of recognizing equivalent expressions, and is commonly used in many of computer algebra systems. Also, rationalizing expression has applications in calculus [2]. More importantly, by rationalizing the denominator, mathematical operations and practical problems can be approximated more accurately [3]. Thus, there are circumstances in which it is advantageous to rationalize an expression.
The general method of denominator rationalization is to seek the rationalized factor of the denominator first and then multiply the numerator and denominator by this factor at the same time [4]. Using this method, the denominator rationalization of quadratic radical fraction has been solved. However, it is difficult to find the rationalized factor for an expression which contains a root higher than a square root [5]. Therefore, a lot of literatures tend to deal with some special case of this problem. In 1929, Paradiso [6] showed that theoretically, in all cases, and practically, in many cases, a rationalizing factor may be found by the method of undetermined coefficients. In 1970, Fateman presented an algorithm named RADCAN that is implemented in MACSYMA for the simplification of expressions containing radicals [7]. Zhou rationalized the denominator for a class of fractions by theory of minimal polynomials in 1986, where is a polynomial whose coefficient are rational and is a complex root of a nonzero rational polynomial [8]. In 1989, Ma showed the possibility of denominator’s rationalization of the irrational expressions: , where is the combination of radical and rational addition, subtraction, and multiplication and is an algebraic element in the field of rational numbers [9]. In 2000, Liu rationalized the denominator for a class of algebraic fractions as follows (where ) using the knowledge of determinants [10]. In 2002, using polynomials, Tang also discussed the algebraic fractions whose denominator is the same with the above fraction [11]. And in 2015, Berele and Catoiu produced an exact formula for rationalizing any fraction whose denominator is a linear combination with rational coefficients of square roots of rational numbers [1].
Existing results mentioned above mainly deal with several kinds of denominators rationalization of radical fractions by using related theory of polynomials. And many other kinds of denominators of irrational fraction are unsolved, such as . This kind of fraction is a very common form in mathematical calculation, so it is of high application value to study a general method to rationalize the denominator. In this paper, we will investigate this by the theory of Gröbner bases as it is an important tool to solve many problems in polynomial ideal [12, 13]. And it has been implemented in many computational softwares including Singular, Maple, CoCoA, Mathematica, Macaulay 2, etc. Many fundamental problems in commutative algebra, computational algebraic number theory, algebraic geometry, graph theory, image processing, cryptography and encoding, and science and engineering can be solved by it algorithmically [13–23]. And the minimal polynomial can be obtained by the reduced Gröbner basis algorithm easily. Based on this and results mentioned above, we consider using the theory of Gröbner bases to explore the relationship between minimal polynomials and denominator rationalization and discuss mainly denominator rationalization of the fraction with the form aswhere . We hope to establish a simplified method for rationalizing the denominator of this type fraction.
The rest of the paper is organized as follows. We present some preliminary knowledge, basic concepts, and a special method to rationalize the denominators with square root and cube root in Section 2. Main results on rationalizing the denominator of a type fraction are shown in Section 3. A simplified algorithm and an example established to illustrate the algorithm are given in Section 4. Section 5 concludes the paper.
2. Denominators with Square Root and Cube Root
In what follows, will denote the polynomial ring in variables with coefficients in a field , will be an algebraic extension of the field , and will denote the extension times of the extension field . will denote the rational number field, will be the set of integers, and will denote the least common multiple of and . For a nonzero polynomial , we use , , and to denote the leading term, the leading coefficient, and the leading monomial of , respectively. For a set , we denote . Then, we introduce several related definitions and algorithm.
Definition 1. Let be an ideal in . A finite subset of is called a Gröbner basis of if
Definition 2 (see [13]). Let , and , and the S-polynomial of and is defined as (Algorithm 1)This section focuses on the problem of rationalizing the denominator with square root and cube root. For the sake of convenience in researching the problem, we put the coefficient in the radical sign, and then, the fraction can be reduced to one of the following forms:We first consider the denominators rationalization of form (1).
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Lemma 1 (see [13]). Let be a subset of an ideal , and then, is a Gröbner bases for if and only if , for all .
Theorem 1. Let be an ideal in , and then, .
Proof. If , then there exists , such that , that is, .
In turn, if , then there exists and , such that , that is, .
Next, we introduce how to rationalize the denominator of fraction as form (1). First, consider the following ideal:where denotes the polynomial ring in variables and on and is an extension of the field with as a variable. LetIn the following, multivariate polynomials are denoted as for convenience. And the term order of the multivariate polynomial ring is the lexicographical ordering, which is defined by .
Note thatBy Lemma 1, we have that is a Gröbner basis of . Let, and then, , which is the denominator we want to deal with. Replace by in Theorem 1, and then, the condition is equivalent to , where is the polynomial we need.
So, the key issue is whether , that is, whether . If true, we can rationalize the denominator of by using this method.
Now, we use Buchberger algorithm to obtain the Gröbner basis of and then determine whether .
Initialize , and using the algorithm, we get the Gröbner basis of iswhereObserve , and it is easy to find that is independent of and , and thus, . Note that is a field, and thus, . Consequently, . Therefore, we can calculate the rationalizing result of .
Then, express as the combination of :that is,Hence, divide both sides of equation (13) by , and substitute the value of and into the equation, and then, we obtainUsing Matlab to simplify the equation above, we obtainFor form (2), we just need to change to , and the other steps are the same.
3. Denominators with Higher Radical
In this section, we focus on rationalizing the denominator of general form aswhere .
By the method in Section 2, we first construct the ideal , where . Then, compute the Gröbner basis and the reduced Gröbner basis of . If , then calculate . Finally, express as a combination of .
However, there are two important uncertainties, one is whether and the other is whether is a combination of . So, this method can only solve the case when are specific values. Because of this limitation, we hope to find another method for the general form.
First, we introduce several related definitions and lemmas.
Definition 3 (see [24]). Let be an algebraic extension of the field , . Suppose is n K-insertion of , where is the complex field. For , defineas the trace of to the expansion .
Definition 4 (see [24]). Let be an algebraic extension of the field . Suppose is the K-insertion of and , and we defineas the discriminant of for the expansion .
Definition 5. Let be a field, , and be a polynomial. We call is the minimal polynomial of in if satisfies the following:(1) is monic and (2)If and , then
Lemma 2 (see [24]). Let be the expansion of number field and . Suppose is the minimal polynomial of in K, where , and then, .
Lemma 3 (see [23]). Let be the discriminant of element for the expansion , and then, if and only if is K-linearly independent.
Lemma 4 (see [25], Eisenstein criterion). Let be an integral coefficient univariate polynomial. If there is a prime number such that(1)(2)(3)
Then, is irreducible over the rational number field .
Using the lemmas above, we can prove the following results.
Theorem 2. If is prime number, then is the minimal polynomial for in rational number filed , where and .
Proof. It is straightforward that is a root of . In the following, we prove that is irreducible.
Let , where . It is obvious that , , and . In fact, we can prove that . Suppose , and then, . Combining , we have that , that is, , and this is a contradiction. So, . By the Eisenstein criterion, is irreducible, so is irreducible, and it is the minimal polynomial for in .
Theorem 3. If are different prime numbers, thenis an irrational number.
Proof. It is straightforward that is a real number. In the following, we prove that it is an irrational number. Suppose it is a rational number, and set . Then, is a positive rational root of . Note that is an integral coefficient polynomial and , and then, the positive rational root of must be a factor of its constant term. So, it has the form as , where , . Then, we have thatFrom the equation above, we see that some of must be 0. Without loss of generality, we denote the elements whose power exponents are nonzero as , and equation (20) turns into the following:that is,Multiply both sides of equation (22) by the power, and then,where . Hence, , and this contradicts are different prime numbers. So, the conclusion is correct.
Theorem 4. If are different prime numbers and are positive integers and no less than 2, then the finite extension times of is no more than .
Proof. Let , and , and then, , that is, can be viewed as the single extension of . So, , where denotes the degree of the polynomial , and is the minimal polynomial of in the field . Obviously, is a root for , . By the definition of minimal polynomial, we see that , . From Theorem 3, we see that is an irreducible polynomial in . By the property of domain extension, we have . Hence,
Theorem 5. If are different prime numbers and are positive integers and no less than 2, then is the minimal polynomial of in the field .
Proof. Let , and in the following, we prove that all the elements in setare linearly independent. First, sort the elements in . It is easy to observe that each element in corresponds to such an array . So, we can turn the problem into sorting the exponents , where the term order is lexicographic order. We denote the element that corresponds to the largest exponential as , the element that corresponds to the second largest exponential as , and the element that corresponds to the smallest exponential as , and then, we have sorted out all the elements in .
Now, we calculate det . Suppose and . If one of the following two situations is true,(1)(2)We obtain that by Theorem 3. Hence, , where . Otherwise, there exists does not satisfy either of the above two cases, and we can calculate by Lemma 2. We first want to obtain the minimal polynomial of in rational number field . SetObviously, . Let and , and by Theorem 3, we have that , for any , where . Hence, is the smallest element in .
In the following, we prove that is the minimal polynomial of in the rational number field .
It is straightforward that . We factorize in the complex field aswhere .
Suppose is reducible in , and then, some constant terms of the linear factor in the decomposition above are rational numbers, that is, there is a and such that , i.e., . So, , that is, . Hence, , and there is an such that . Note that , and then, , and this contradicts the selection of . Thus, is irreducible in , and then, is the minimal polynomial of in rational number field . Combined with Lemma 2, we see that .
Based on the discussion above, we obtain that . Then, all elements in set are linearly independent in by Lemma 3. Combined with Theorem 4, we see thatSo, is the minimal polynomial of in the field .
Based on the results above, we can present the steps for rationalizing denominators of the following form:where are integers and and . Step 1: write as fractions in the lowest term, . Then, decompose into power product of different prime factors, respectively. We use to denote the common prime factors of and , to denote the different prime factors in , and to denote the different prime factors in . Step 2: let , andWe construct the rational extension field:Let , and there are such thatAccording to Theorem 2, we have that is the minimal polynomial of in the rational number field . It is also known, by Theorem 5, is the minimal polynomial of in for , is the minimal polynomial of in for , and is the minimal polynomial of in for .
Next, we construct the following homomorphic maps:For any , where . DefineUnder this map, the corresponding preimage of is itself, where . Step 3: we find the minimal polynomial of in according to [26].The term order in is the lexicographical ordering defined by . Under this term order, we calculate the reduced Gröbner basis of the idealwhereThen, we compute and takeHence, is the minimal polynomial of in . And then we can rationalize the denominator by applying the minimal polynomial.
Set ( is a parameter) and divide by such thatThen, and , otherwise, it contradicts that contains . Correspondingly, equation (38) can be rewritten asSubstitute into the equation above, and then, , that is,which is the desired result of rationalizing denominators.
4. Algorithm and Example
According to the theorems and discussion in Section 3, we obtain an algorithm for rationalizing the denominators of fractions with the form as . We describe this algorithm in more detail in Figure 1.
In the following, we construct an example to show the effectiveness of the algorithm.
Example 1. Rationalizing the denominator of Step 1: we know a = 6 and b = 4, and the command “format rat” can be omitted here. Carry out the prime factorization of 6 and 4 by using the function “factorization,” and we obtainIt is easy to see that is a common prime factor of 6 and 4. In addition, also contains the factor . Step 2: note that , and then, using the function “min_GBS,” we obtainLetWe construct the extension field of the rational number field . Set , and then, . Step 3: hence, is a minimal polynomial of on and is a minimal polynomial of on .Settake the lexicographical ordering as the term order in , and calculate the reduced Gröbner basis of ideal in the software singular using the built-in function “groebner.” Then, the calculation results are as follows:whereTherefore, . Hence, the minimal polynomial of in the rational number field is .
Next, divide by in the software Matlab using the built-in function “polynomialReduce.” Then,whereSubstitute intothen
Remark 1. Note that , and we can use the method in Section 2 to do this example. Replace and do all the steps as in Section 2, and we obtainThrough the two examples above, it is easy to find that the results obtained by both methods are the same, which proves that the algorithm we designed is correct and the application scope is wider.
Remark 2. Using the method in Section 2 or the method of undetermined coefficients, the problem of denominator rationalization of fractions with the form as may not be solved or can be solved but will take a long time. However, from Example 1, it is straightforward that this problem can be solved efficiently and simply according to the three steps of Denominator rationalized algorithm by using the software Matlab and Singular.
5. Conclusion
In this paper, using theory of Gröbner bases, we have achieved in rationalizing denominators for two types of fractions, especially rationalizing the denominator of the fraction with the form aswhere . We have presented some interesting results and an executable algorithm on rationalizing the denominators for this type fractions. Furthermore, we have established an example to illustrate the effectiveness of the algorithm.
The method that we proposed on rationalizing denominators can be realized in computer system such as Maple and Singular, which makes the related computation more quick. Furthermore, it can improve the performance of the related algorithms. For example, we find that, for some new swarm intelligence algorithms proposed in recent years, such as monarch butterfly optimization (MBO) [27], earthworm optimization algorithm (EWA) [28], elephant herding optimization (EHO) [29], etc. the fraction of the form occurs during the implementation of these algorithms. The denominator rationalized algorithm we proposed can rationalize the denominator of this type fraction, which can improve the accuracy of these algorithms on calculation.
Future work will investigate in rationalizing denominators of more general forms of radical fraction. The type of fractions discussed in the paper is unnested radical expressions, and the problem of the nested radical fractions is not involved here, which is also what we will do.
Data Availability
Data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (11871207 and 11971161).