Dade's Invariant Conjecture for the Symplectic Group Sp4(2 n ) and the Special Unitary Group SU4(22n ) in Defining Characteristic

In this article, we verify Dade's projective invariant conjecture for the symplectic group Sp₄(2 ⁿ ) and the special unitary group SU₄(2²ⁿ ) in the defining characteristic, that is, in characteristic 2. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for Sp₄(2 ⁿ ) and SU₄(2²ⁿ ) in the defining characteristic, that is, Sp₄(2 ⁿ ) and SU₄(2²ⁿ ) are good for the prime 2 in the sense of Isaacs, Malle, and Navarro.     

Polynomials and primitive roots in finite fields

The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k. If f(x) is a polynomial over k of small degree compared to the size of k, then f(x) represents at least one primitive element of k. Also f(x) represents an lth power at a primitive element of k, if l is also small. As a consequence of this, the following results holds.Theorem. Let g(x) be a square-free polynomial with integer coefficients. For all but finitely many prime numbers p, there is an integer a such that g(a) is equivalent to a primitive element modulo p.Theorem. Let l be a fixed prime number and f(x) be a square-free polynomial with integer coefficients with a non-zero constant term. For all but finitely many primes p, there exist integers a and b such that a is a primitive element and f(a) ≡ b¹ modulo p.     

Some Remarks on Syzygies of Adjoint Line Bundles

In this article, we construct examples of n-folds X carrying an ample line bundle A ∈ Pic X such that property N _p fails for K _X  + (n + 1 + p)A. This shows that the condition of Mukai's conjecture is optimal for every n ≥ 1 and p ≥ 0.     

Coefficients of ternary cyclotomic polynomials

It is customary to define a cyclotomic polynomial Φn(x) to be ternary if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn(x) and M(p) be the maximum of A(pqr). In 1968, Sister Marion Beiter (1968, 1971) [3] and [4] conjectured that . In 2008, Yves Gallot and Pieter Moree (2009) [6] showed that the conjecture is false for every p≥11, and they proposed the Corrected Beiter conjecture: . Here we will give a sufficient condition for the Corrected Beiter conjecture and prove it when p=7.     

Mean value conjectures for rational maps

Let p be a polynomial in one complex variable. Smale's mean value conjecture estimates |p′(z)| in terms of the gradient of a chord from (z,?p(z)) to some stationary point on the graph of p. The conjecture does not immediately generalize to rational maps since its formulation is invariant under the group of affine maps, not the full Möbius group. Here we give two possible generalizations to rational maps, both of which are Möbius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture. Finally, we discuss some candidate extremal rational maps, namely rational maps all of whose critical points are fixed points.     

On the number of equivalence classes of attracting dynamical systems

We study discrete dynamical systems of the kind h(x) = x + g(x), where g(x) is amonic irreducible polynomial with coefficients in the ring of integers of a p-adic field K. The dynamical systems of this kind, having attracting fixed points, can in a natural way be divided into equivalence classes, and we investigate whether something can be said about the number of those equivalence classes, for a certain degree of the polynomial g(x). The text was submitted by the authors in English.     

On roots of exponential terms

In the present paper some tools are given to state the exact number of roots for some simple classes of exponential terms (with one variable). The result were obtained by generalizing Sturm's technique for real closed fields. Moreover for arbitrary non-zero terms t(x) certain estimations concerning the location of roots of t(x) are given. MSC: 03C65, 03C60, 12L12.     

Zero distribution and division results for exponential polynomials

An exponential polynomial of order q is an entire function of the form

$$g(z) = {P_1}(z){e^{{Q_1}(z)}} + ...{P_k}(z){e^{{Q_k}(z)}},$$

where the coefficients P_j(z),Q_j(z) are polynomials in z such that

$$\max \{ deg({Q_j})\} = q.$$

It is known that the majority of the zeros of a given exponential polynomial are in domains surrounding finitely many critical rays. The shape of these domains is refined by showing that in many cases the domains can approach the critical rays asymptotically. Further, it is known that the zeros of an exponential polynomial are always of bounded multiplicity. A new sufficient condition for the majority of zeros to be simple is found. Finally, a division result for a quotient of two exponential polynomials is proved, generalizing a 1929 result by Ritt in the case q = 1 with constant coefficients. Ritt’s result is closely related to Shapiro’s conjecture that has remained open since 1958.

     

Improving Thompson's Conjecture for Suzuki Groups

Let G be a finite group and cs(G) be the set of conjugacy class sizes of G. In 1987, J. G. Thompson conjectured that, if G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying that cs(G) = cs(M), then G ? M. This conjecture has been proved for Suzuki groups in [5 Guiyun, C. (1996). On Thompson's conjecture. J. Algebra 185(1):184–193.[Crossref], [Web of Science ®] , [Google Scholar]]. In this article, we improve this result by proving that, if G is a finite group such that cs(G) = cs(Sz(q)), for q = 2^2m+1, then G ? Sz(q) × A, where A is abelian. We avoid using classification of finite simple groups in our proofs.     

Nested Log-Concavity

We give conditions on the coefficients of a polynomial p(x) so that p(x + t) be log-concave or strictly log-concave. Several applications are given: if p(x) is a polynomial with nonnegative and nondecreasing coefficients, then p(x + t) is strictly log-concave for all t ≥ 1; for any polynomial p(x) with positive leading coefficient, there is t ₀ ≥ 0 such that for any t ≥ t ₀ it holds that the coefficients of p(x + t) are positive, strictly decreasing, and strictly log-concave; if p(x) is a log-concave polynomial with nonnegative coefficients and no internal zeros, then p(x + t) is strictly log-concave for all t > 0; Betti numbers of lexsegment monomial ideals are strictly log-concave.     

Uniform Boundedness of Torsion Subgroups of Linear Groups

The torsion conjecture says： for any abelian variety A defined over a number field k, the order of the torsion subgroup of A（k） is bounded by a constant C（k,d） which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G（k）. Then the order of Г is bounded by a constant C＇（k, d） which depends only on k and the dimension d of G.     

Thomassen's conjecture implies polynomiality of 1‐Hamilton‐connectedness in line graphs

A graph G is 1‐Hamilton‐connected if G?x is Hamilton‐connected for every x∈V(G), and G is 2‐edge‐Hamilton‐connected if the graph G+ X has a hamiltonian cycle containing all edges of X for any X?E⁺(G) = {xy| x, y∈V(G)} with 1≤|X|≤2. We prove that Thomassen's conjecture (every 4‐connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4‐connected line graph is 1‐Hamilton‐connected and/or 2‐edge‐Hamilton‐connected. As a corollary, we obtain that Thomassen's conjecture implies polynomiality of both 1‐Hamilton‐connectedness and 2‐edge‐Hamilton‐connectedness in line graphs. Consequently, proving that 1‐Hamilton‐connectedness is NP‐complete in line graphs would disprove Thomassen's conjecture, unless P = NP. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 241–250, 2012     

Graham's pebbling conjecture on products of cycles

Chung defined a pebbling move on a graph G to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graph is the smallest number f(G) such that any distribution of f(G) pebbles on G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G and H, f(G×H)≤ f(G)f(H). We prove Graham's conjecture when G is a cycle for a variety of graphs H, including all cycles. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 141–154, 2003     

Alternating sums over π-subgroups

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace p by a set of primes π and prove a π-version of Dade's conjecture for π-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a π-version of Alperin's weight conjecture previously established by the authors.     

Some Consequences of Schanuel's Conjecture in Exponential Rings

In last years Schanuel's Conjecture has played a fundamental role in Transcendental Number Theory and in decidability issues.

In this article we investigate algebraic relations among the elements of the exponential field (?, e ^x ) modulo Schanuel's Conjecture. We prove that there are no further relations between π and i assuming Schanuel's Conjecture except the known ones, e ^πi  = ?1 and i ² = ?1. Moreover, modulo Schanuel's Conjecture we prove that the E-subring of ? generated by π is isomorphic to the free exponential ring on π.     

List‐coloring the square of a subcubic graph

The square G² of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G²) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list‐chromatic number of G² equals the chromatic number of G², that is, χ_l(G²) = χ(G²) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χ_l(G²) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χ_l(G²) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χ_l(G²) ≤ 7. Dvo?ák, ?krekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χ_l(G²) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χ_l(G²) ≤ 6. All of our proofs can be easily translated into linear‐time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

Oscillation criteria for fourth‐order differential equations with middle term

Oscillation criteria for self‐adjoint fourth‐order differential equations were established for various conditions on the coefficients r(x) > 0, q(x) and p(x). However, most of these results deal with the case when lim_{x → ∞}∫^x₁q(s) ds < +∞. In this note we give a new oscillation criterion in the case when this condition is not fulfilled, in particular when q(x)↗ + ∞ (even with exponential growth).     

Some Results on the Linearized Oscillation of the Odd-order Neutral Difference Equation

In this article, we obtain some results on the linearized oscillation of the odd-order neutral difference equation <artwork name="GAPA31013eu1"> where Δ is the forward difference operator, m is odd, {p_n },{q_n } are sequences of nonnegative real numbers, k, l are nonnegative integers, g(x),h(x)∈ C(R,R) with xg(x) > 0 for x ≠0.