# Polynomials with small value set over finite fields

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### Abstract

Let Kq denote the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coeficients in Kq. Let C(f) denote the number of distinct values of f(x) as x ranges over Kq. We easily see that C(f)≥ q-1 d+1 where [{norm of matrix}] is the greatest integer function. A polynomial for which equality in (*) occurs is called a minimum value set polynomial. There is a complete characterization of minimum value set polynomials over arbitrary finite fields with d < √q and C(f) ≥ 3 [see L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Strauss, Mathematika 8 (1961), 121-130; W. H. Mills, Pacific J. Math. 14 (1964), 225-241]. In this paper we give a complete list of polynomials of degree d4 < q which have a value set of size less than 2q d, twice the minimum possible. If d > 15 then f(x) is one of the following polynomial forms: 1. (a) f(x) = (x + a)d + b, where d | (q - 1), 2. (b) f(x) = ((x + a) d 2 + b)2 + c, where d | (q2 - 1), 3. (c) f(x) = ((x + a)2 + b) d 2 + c, where d | (q2 - 1), or 4. (d) f(x) = Dd,a(x + b) + c, where Dd,a(x) is the Dickson polynomial of degree d, d | (q2 - 1) and a is a 2kth power in Kq2 where d = 2kr, r is odd. The result is obtained by noticing the connection between the size of the value set of a polynomial f(x) and the factorization of the associated substitution polynomial f*(x, y) = f(x) - f(y) in the ring Kq[x, y]. Essentually, we show that C(f) < 2q d implies that f*(x, y) has at least d 2 factors in Kq[x, y], and we determine all the polynomials with such characteristic.

Original language English (US) 167-188 22 Journal of Number Theory 28 2 https://doi.org/10.1016/0022-314X(88)90064-9 Published - 1988

### Fingerprint

Galois field
Polynomial
Dickson Polynomials
Denote
Monic polynomial
Substitution
Factorization
Equality
Odd
Distinct
Norm
Ring
Imply
Arbitrary
Range of data

### ASJC Scopus subject areas

• Algebra and Number Theory

### Cite this

Polynomials with small value set over finite fields. / Gomez-Calderon, Javier; Madden, Daniel.

In: Journal of Number Theory, Vol. 28, No. 2, 1988, p. 167-188.

Research output: Contribution to journalArticle

Gomez-Calderon, Javier ; Madden, Daniel. / Polynomials with small value set over finite fields. In: Journal of Number Theory. 1988 ; Vol. 28, No. 2. pp. 167-188.
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abstract = "Let Kq denote the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coeficients in Kq. Let C(f) denote the number of distinct values of f(x) as x ranges over Kq. We easily see that C(f)≥ q-1 d+1 where [{norm of matrix}] is the greatest integer function. A polynomial for which equality in (*) occurs is called a minimum value set polynomial. There is a complete characterization of minimum value set polynomials over arbitrary finite fields with d < √q and C(f) ≥ 3 [see L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Strauss, Mathematika 8 (1961), 121-130; W. H. Mills, Pacific J. Math. 14 (1964), 225-241]. In this paper we give a complete list of polynomials of degree d4 < q which have a value set of size less than 2q d, twice the minimum possible. If d > 15 then f(x) is one of the following polynomial forms: 1. (a) f(x) = (x + a)d + b, where d | (q - 1), 2. (b) f(x) = ((x + a) d 2 + b)2 + c, where d | (q2 - 1), 3. (c) f(x) = ((x + a)2 + b) d 2 + c, where d | (q2 - 1), or 4. (d) f(x) = Dd,a(x + b) + c, where Dd,a(x) is the Dickson polynomial of degree d, d | (q2 - 1) and a is a 2kth power in Kq2 where d = 2kr, r is odd. The result is obtained by noticing the connection between the size of the value set of a polynomial f(x) and the factorization of the associated substitution polynomial f*(x, y) = f(x) - f(y) in the ring Kq[x, y]. Essentually, we show that C(f) < 2q d implies that f*(x, y) has at least d 2 factors in Kq[x, y], and we determine all the polynomials with such characteristic.",
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N2 - Let Kq denote the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coeficients in Kq. Let C(f) denote the number of distinct values of f(x) as x ranges over Kq. We easily see that C(f)≥ q-1 d+1 where [{norm of matrix}] is the greatest integer function. A polynomial for which equality in (*) occurs is called a minimum value set polynomial. There is a complete characterization of minimum value set polynomials over arbitrary finite fields with d < √q and C(f) ≥ 3 [see L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Strauss, Mathematika 8 (1961), 121-130; W. H. Mills, Pacific J. Math. 14 (1964), 225-241]. In this paper we give a complete list of polynomials of degree d4 < q which have a value set of size less than 2q d, twice the minimum possible. If d > 15 then f(x) is one of the following polynomial forms: 1. (a) f(x) = (x + a)d + b, where d | (q - 1), 2. (b) f(x) = ((x + a) d 2 + b)2 + c, where d | (q2 - 1), 3. (c) f(x) = ((x + a)2 + b) d 2 + c, where d | (q2 - 1), or 4. (d) f(x) = Dd,a(x + b) + c, where Dd,a(x) is the Dickson polynomial of degree d, d | (q2 - 1) and a is a 2kth power in Kq2 where d = 2kr, r is odd. The result is obtained by noticing the connection between the size of the value set of a polynomial f(x) and the factorization of the associated substitution polynomial f*(x, y) = f(x) - f(y) in the ring Kq[x, y]. Essentually, we show that C(f) < 2q d implies that f*(x, y) has at least d 2 factors in Kq[x, y], and we determine all the polynomials with such characteristic.

AB - Let Kq denote the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coeficients in Kq. Let C(f) denote the number of distinct values of f(x) as x ranges over Kq. We easily see that C(f)≥ q-1 d+1 where [{norm of matrix}] is the greatest integer function. A polynomial for which equality in (*) occurs is called a minimum value set polynomial. There is a complete characterization of minimum value set polynomials over arbitrary finite fields with d < √q and C(f) ≥ 3 [see L. Carlitz, D. J. Lewis, W. H. Mills, and E. G. Strauss, Mathematika 8 (1961), 121-130; W. H. Mills, Pacific J. Math. 14 (1964), 225-241]. In this paper we give a complete list of polynomials of degree d4 < q which have a value set of size less than 2q d, twice the minimum possible. If d > 15 then f(x) is one of the following polynomial forms: 1. (a) f(x) = (x + a)d + b, where d | (q - 1), 2. (b) f(x) = ((x + a) d 2 + b)2 + c, where d | (q2 - 1), 3. (c) f(x) = ((x + a)2 + b) d 2 + c, where d | (q2 - 1), or 4. (d) f(x) = Dd,a(x + b) + c, where Dd,a(x) is the Dickson polynomial of degree d, d | (q2 - 1) and a is a 2kth power in Kq2 where d = 2kr, r is odd. The result is obtained by noticing the connection between the size of the value set of a polynomial f(x) and the factorization of the associated substitution polynomial f*(x, y) = f(x) - f(y) in the ring Kq[x, y]. Essentually, we show that C(f) < 2q d implies that f*(x, y) has at least d 2 factors in Kq[x, y], and we determine all the polynomials with such characteristic.

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