International Journal of Mathematics and Mathematical Sciences

Volume 2016, Article ID 6093219, 7 pages

http://dx.doi.org/10.1155/2016/6093219

## Characterization and Enumeration of Good Punctured Polynomials over Finite Fields

^{1}Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand^{2}Department of Mathematics, Mahidol Wittayanusorn School, Nakhon Pathom 73170, Thailand

Received 25 November 2015; Revised 3 March 2016; Accepted 6 March 2016

Academic Editor: Shyam L. Kalla

Copyright © 2016 Somphong Jitman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.