Difference sets and the irreducibles in function fields

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 ,. .. , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts that V(d, s, a) = µ d q + O(1), where V(d, s, a) is such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 ,. .. , d d−s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q-rational points is established.

Journal of Combinatorial Theory, Series A, 2014

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 ,. .. , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts that V(d, s, a) = µ d q + O(1), where V(d, s, a) is such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 ,. .. , d d−s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q-rational points is established.

Acta Arithmetica, 2014

arXiv (Cornell University), 2015

We estimate the average cardinality V(A) of the value set of a general family A of monic univariate polynomials of degree d with coefficients in the finite field Fq. We establish conditions on the family A under which V(A) = µ d q + O(q 1/2), where µ d := d r=1 (−1) r−1 /r!. The result holds without any restriction on the characteristic of Fq and provides an explicit expression for the constant underlying the O-notation in terms of d. We reduce the question to estimating the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of Fq-rational points.

Canadian Journal of Mathematics, 1969

If A is a set with only a finite number of elements, we write |A| for the number of elements in A. Let p be a large prime and let m be a positive integer fixed independently of p. We write [pm ] for the finite field with pm elements and [pm ]′ for [pm ] – {0}. We consider in this paper only subsets H of [pm ] for which |H| = h satisfies 1.1 If f(x) ∈ [pm, x] we let N(f; H) denote the number of distinct values of y in H for which at least one of the roots of f(x) = y is in [pm ]. We write d(d ≥ 1) for the degree of f and suppose throughout that d is fixed and that p ≧ p 0(d), for some prime p 0, depending only on d, which is greater than d.

Bulletin of the London Mathematical Society, 1985

Finite Fields and Their Applications, 2012

We show that, for any integer ℓ with q − √ p − 1 ≤ ℓ < q − 3 where q = p n and p > 9, there exists a multiset M satisfying that 0 ∈ M has the highest multiplicity ℓ and b∈M b = 0 such that every polynomial over finite fields Fq with the prescribed range M has degree greater than ℓ. This implies that Conjecture 5.1. in [1] is false over finite field Fq for p > 9 and k := q − ℓ − 1 ≥ 3.

Finite Fields and Their Applications, 2011

In this paper we study the relation between coefficients of a polynomial over finite field F q and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m | q − 1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m q−1 and m ≤ q−1 2. As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q 3/2) operations.

arXiv (Cornell University), 2023

We study the preorder ≤ p on the family of subsets of an algebraically closed field of characteristic 0 defined by letting A ≤ p B if there exists a polynomial P such that A = P −1 (B).

Mathematics of Computation, 2012

P. Turán asked if there exists an absolute constant C such that for every polynomial f ∈ Z[x] there exists an irreducible polynomial g ∈ Z[x] with deg(g) ≤ deg(f ) and L(f − g) ≤ C, where L(·) denotes the sum of the absolute values of the coefficients. We show that C = 5 suffices for all integer polynomials of degree at most 40 by investigating analogous questions in Fp[x]

Mathematika, 2000

Let E be a local field, i.e., a field which is complete with respect to a rank one discrete valuation υ (we do not require any finiteness condition on the residue class field of E). Let f(X) be a polynomial in one variable, with coefficients in E. It is well known [4, 6, 9, 11, 13] that the Newton polygon method allows us to gather information about the factorization of f(X). This method consists of attaching to each side S of a Newton polygon of f(X) a factor (not necessarily irreducible) of f(X), the degree of which is the length of the horizontal projection of S.

2003

In this note we study the following question: Under which minimal assumptions on the function f is it true that the condition (∆ N +1 h f)(a) = 0 for all a ∈ R and h ≥ 0 implies that f is a polynomial of degree less than or equal to N ?

European Journal of Combinatorics, 1990

We present a condition on the intersection numbers of difference sets which follows from a result of Jungnickel and Pott [3]. We apply this condition to. rule out several putative (non-abelian) difference sets and to correct erroneous proofs of Lander [4] for the nonexistence of (352, 27, 2)-and (122, 37, 12)-difference sets.

Combinatorica, 1991

In this paper, we shall prove several non-existeuce results for divisible difference sets, using three approaches: (i) character sum arguments similar to the work of Turyn [25] for ordinary difference sets, (ii) involution arguments and (iii) rrrnltipliers in conjunction with results on ordinary difference sets. Among other results, we show that an abclian affine diffcren<:c set of odd orders (snot a perfect square) in G can exist only if the Sylow 2-subgroup of G is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd order n.

Discrete Mathematics, 1999

Van Lint and MacWilliams (IEEE Trans. Inform. Theory IT 24 (1978) 730-737) conjectured that the only q-subset X of GF(q 2), with the properties 0; 1 ∈ X and x − y is a square for all x; y ∈ X , is the set GF(q). Aart Blokhuis (Indag. Math. 46 (1984) 369-372) proved the conjecture for arbitrary odd q. In this paper we give a similar characterization of GF(q) in GF(q 2), proving the analogue of Blokhuis' theorem for dth powers (instead of squares), when d|(q + 1). We also prove an embedding-type result, stating that if |S| ¿ q − (1 − 1=d) √ q with the same properties as X above, then S ⊆ GF(q).