Pingzhi Yuan

Permutation polynomials over finite fields from a powerful lemma

Abstract Using a lemma proved by Akbary, Ghioca, and Wang, we derive several theorems on permutation polynomials over finite fields. These theorems give not only a unified treatment of some earlier constructions of permutation polynomials, but also new specific permutation polynomials over F q . A number of earlier theorems and constructions of permutation polynomials are generalized. The results presented in this paper demonstrate the power of this lemma when it is employed together with other techniques.

Permutation Trinomials Over Finite Fields with Even Characteristic

Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.

The set of distances in Krull monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property: for each a ∈ H and each two factorizations z, z of a, there exist factorizations z = z0, . . . , zk = z ′ of a such that, for each i ∈ [1, k], zi arises from zi−1 by replacing at most N atoms from zi−1 by at most N new atoms. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H) ≤ 4.

On the rationality of Cantor and Ahmes series

Abstract We give criteria for the rationality of Cantor series and series where a1, a2, ··· and b1, b2, ··· are integers such that an > 0 and the series converge. We precisely say when is rational (i) if {an}n=1∞ is a monotonic sequence of integers and bn + 1 − bn = o(an + 1) or lim and (ii) if for all large n. We give similar criteria for the rationality of Ahmes series and more generally series . For example, if bn > 0 and lim , where An = lcm(a1, a2, ···, an), then is rational if and only if for large n. On the other hand, we show that such results are impossible without growth restrictions. For example, we show that for any integers d > c > 1 there is a sequence {bn}n = 1∞ such that every number x from some interval can be represented as with an ∈ {c,d} for all n.

On algebraic approximations of certain algebraic numbers

Abstract Using hypergeometric functions and the Thue–Siegel method we give an effective improvement of Liouvilles approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalities |BX 4 −AX 3 Y−6BX 2 Y 2 +AXY 3 +BY 4 |⩽N for A⩾58|B|3 or A⩾308B4.

On the Diophantine equation ax2+by2=ckn

In this paper we investigate the solvability and the representation of the solutions of the equation ax2 +by2 = ckn. We extend and improve many known results. In particular, we completely solve the equation (a ± 1)x2 + (3a Ŧ 1) = 4an, 2 ∤ n.

Permutation polynomials from piecewise functions

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we present some classes of explicit permutation polynomials of the forms similar to ( x p k + a x + ? ) p n - 1 d + 1 - a x , d = 2 , 3 , 4 , 6 . The results here generalize the similar results obtained in 5,15 greatly.

Permutation trinomials over F2m

Abstract Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over F 2 m in Zieves paper [30] . We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over F 2 m . Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.

Rational and algebraic approximations of algebraic numbers and their application

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and ε =x0 denotes the fundamental solution ofx2−Dy2=−1, thenx2+1=Dy4 is solvable if and only ify0=A2, where A is an integer. Moreover, if ε>64, thenx2+1=Dy4 has at most one positive integral solution (x, y).

On *-clean group rings

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vas asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.