Abstract
A family of good punctured polynomials is introduced. The complete characterization and enumeration of such polynomials are given over the binary field . Over a nonbinary finite field , the set of good punctured polynomials of degree less than or equal to are completely determined. For , constructive lower bounds of the number of good punctured polynomials of degree over are given.
1. Introduction
From the fundamental theorem of algebra, every polynomial over the rational numbers (or over the real numbers ) has a root in . However, it is not guaranteed that a polynomial has a root in or in . Therefore, for a given polynomial over (resp., ), it is of natural interest to determine whether it has a root in (resp., ). In general, determining whether a given polynomial has a root in a nonalgebraically closed field is an interesting problem and has been extensively studied (see, e.g., [1–4]).
In this paper, we introduce punctured forms of a polynomial over a field (see the definition below) and focus on determining whether the punctured parts of have a root in . Due to the rich algebraic structures and various applications of polynomials over finite fields (see [5–9] and references therein), their properties such as factorization, root finding, and irreducibility have extensively been studied (see [10–13]). In this paper, we mainly focus on punctured polynomials over a finite field which is not an algebraically closed field. The readers may refer to [5] for more details on finite fields and polynomials over finite fields.
Let be a prime power and let denote the finite field of elements. Denote by the set of nonzero elements in . Letbe the set of all polynomials with indeterminate over . Let be the set of all polynomials of degree less than or equal to over and the set of all polynomials of degree over , respectively.
Given a polynomial of degree , for each , the th punctured polynomial of is defined to be
For convenience, by abuse of notation, the degree of zero polynomial is defined to be . Hence, we can write for all constant polynomials .
A polynomial of degree is said to be good punctured if has a root in for all . Otherwise, is said to be bad punctured. The constant polynomials are always good punctured and referred to as trivial good punctured polynomials. A good punctured polynomial is called nontrivial if it is not a trivial good punctured polynomial.
Example 1. Let be a polynomial in . Then , , and . It is not difficult to see that is good punctured.
Given a positive integer and a prime power , let and denote the set of good punctured polynomials of degree less than or equal to over and the set of all good punctured polynomials of degree over , respectively. Precisely, By convention, since , we have and .
Remark 2. From the definitions of and , we have the following facts: (1) is a subset of .(2) is a disjoint union for all .(3) for all .
Example 3. Over the finite fields , we haveHence, and , respectively.
In this paper, we focus on the characterization and enumeration of the good punctured polynomials of degree over . The complete characterization and enumeration of good punctured polynomials over the binary field are given in Section 2. In Section 3, good punctured polynomials of degree over , where , are studied. The good punctured polynomials of degree less than or equal to over fields are completely determined. Lower bounds of the size of the set of good punctured polynomials of degree greater than are provided as well. Conclusion and some discussions about future researches on punctured polynomials are provided in Section 4.
2. Good Punctured Polynomials over the Binary Field
In this section, we focus on good punctured polynomials over the finite field . The characterization and enumeration of such polynomials are completely determined.
First, we determine the set of good punctured polynomials of degree less than or equal to over the binary field . It is not difficult to see that . For , the set is given as follows.
Theorem 4. Let be a positive integer. Then where , , and .
Proof. First, we prove that . Let . We distinguish the proof into three cases.
Case 1 (). Then for some . It follows that is a root of for all . Hence, .
Case 2 (). Then for some . We have and for all . Hence, has a root in for all . Therefore, .
Case 3 (). Then for some . It follows that is a root of and for all . Therefore, has a root in for all . As desired, .
On the other hand, let . Write and consider the following two cases.
Case 1 ()
Case 1.1 (). Then .
Case 1.2 (). Then . Since , we have . It follows that . Hence, .
Case 2 (). Since , we have . Suppose that there exists such that . Since and , we haveIt follows that , a contradiction. Hence, for all . Since , the degree of must be even. We conclude that .
From the two cases, we have , and, hence, .
Therefore, as desired.
Corollary 5. If is a positive integer, then
Proof. By direct calculation, we have and .
Next, assume that . By Theorem 4, we havewhere , , and .
Since and are disjoint, by the inclusion-exclusion principle, we have Clearly, and . Observe that if and only if is even. Hence, It is not difficult to see that and, hence,Therefore, as desired.
Next, we determine the set of good punctured polynomials of degree over the binary field . Since , we have . For , the set can be determined as follows.
Theorem 6. If is a positive integer, then where , , and .
Proof. We prove the statement by determining the elements in of degree . Let , , and be defined as in Theorem 4.
It is not difficult to see that the set of elements in (resp., ) of degree is (resp., ).
If is even, then the set of elements in of degree is . In the case where is odd, the set of elements in of degree is empty.
By Theorem 4, the result, therefore, follows.
Corollary 7. If is a positive integer, then
Proof. By direct calculation, we have andHence, we have and .
Next, assume that . By Theorem 6, we havewhere , , and are defined as in Theorem 6. Since and are disjoint, by the inclusion-exclusion principle, we have We note that and .
Since if and only ifis odd, we have It is not difficult to see that and, hence,Therefore, by (18), we have as desired.
Table 1 presents the numbers and for . The relation in Remark 2 can be easily seen.
3. Punctured Polynomials over Nonbinary Finite Fields
In this section, we focus on punctured polynomials over nonbinary finite fields. Given a prime power , the characterization and enumeration of good punctured polynomials of degree less than or equal to over are completely determined. For , we construct subsets of and which lead to lower bounds of the cardinalities of and , respectively.
Theorem 8. If is a prime power, then .
Proof. By the definition, . Let . Since has a root in , we have . Hence, as desired.
The next corollary follows immediately from Theorem 8.
Corollary 9. If is a prime power, then the following statements hold:(i).(ii).(iii).
Theorem 10. Let be a prime power. Then
Proof. Let and .
Let . We write and consider the proof as two cases.
Case 1 (). We have , , and . Since and are nonzero, it follows that , , and are roots of , , and , respectively. Hence, .
Case 2 (). Then for some . It follows that is a root of , , and . Therefore, .
Case 3 (). Then, by the definition, .
On the other hand, let . If , then , and, hence, by Theorem 8. Assume that . Then , , and have a root in .
Case 1 (). We have that has a root in which implies that . Therefore, .
Case 2 (). Since has a root in , we have . Hence, .
From the two cases, it can be concluded that .
As desired, we have .
Corollary 11. If is a prime power, then
Proof. Let and be defined as in the proof of Theorem 10. It is not difficult to see that , , and are disjoint. By Theorem 10, we have as desired.
Corollary 12. If is a prime power, then
Proof. From Theorem 10, it is not difficult to see that the polynomials of degree less than in are . Hence, the result follows.
Corollary 13. If is a prime power, then
Proof. From Corollaries 11 and 12, it follows that
In the case where , determining the sets and is more tedious and complicated. For these cases, we give constructive lower bounds of and .
The following results are important tools in constructing lower bounds of and .
Theorem 14 (see [14, Page 588]). Let be a positive integer and let be a prime power. Then the number of monic irreducible polynomials of degree in iswhereis the Möbius function.
Theorem 15 (see [3, Section 4.2, Theorem 1]). Let be a polynomial of degree or in . Then is reducible if and only if has a root in .
Theorem 16. If is a prime power, then
Proof. Let .
First, we show that . Let , where and . Then has a root in and is a root of Hence, is good punctured.
By Theorem 14, the number of monic irreducible polynomials of degree over is Hence, the number of irreducible polynomials of degree over isBy Theorem 15, the number of polynomials of degree over having a root in isSince , we haveas desired.
Corollary 17. If is a prime power, then
Proof. By Corollary 11 and Theorem 16, we haveHence, by Remark 2, we have the relation
Theorem 18. If is a prime power, then
Proof. Let .
First, we show that . Let , where and . Then has a root in and is a root of Therefore, is good punctured as desired.
By Theorem 14, the number of monic irreducible polynomials of degree over is Hence, the number of irreducible polynomials of degree over isBy Theorem 15, the number of polynomials of degree over having a root in isSince , we have as desired.
Corollary 19. If is a prime power, then
Proof. By Corollary 17, we haveFrom Theorem 18, we haveHence, by Remark 2, we have the relation
Theorem 20. Let be a prime power and let be an integer. Then
Proof. Let .
First, we show that . Clearly, . Let . Then is a root of for all , and, hence, . Note that . By Corollary 11, we haveTherefore, consider
Corollary 21. Let be a prime power and let be an integer. Then .
Proof. The set of elements in of degree in the proof of Theorem 20 is . By Theorem 20, we have
4. Conclusion and Open Problems
The concepts of punctured polynomials and good punctured polynomials are introduced. Over the finite field , the complete characterization and enumeration of such polynomials are given. Over nonbinary finite fields, the good punctured polynomials of degree less than or equal to are completely determined. For , constructive lower bounds of the number of good punctured polynomials of degree are given.
In general, the following related problems are also interesting:(1)Determine the sets and , where is a prime power and is an integer.(2)Determine the exact values of and , where is a prime power and is an integer.(3)Improve lower bounds of and , where is a prime power and is an integer.(4)Characterize and enumerate the good punctured polynomials of degree over the real numbers or over the rational numbers .
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This research is supported by the Thailand Research Fund under Research Grant TRG5780065.