An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

On p−q=c and related three term exponential Diophantine equations with prime bases

Using a theorem on linear forms in logarithms, we show that the equation p^x−2^y=p^u−2^v has no solutions (p,x,y,u,v) with x≠u, where p is a positive prime and x,y,u, and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 10¹⁵. More generally, we obtain a similar result for p^x−q^y=p^u−q^v>0 where q is a positive prime, . We solve a question of Edgar showing there is at most one solution (x,y) to p^x−q^y=2^h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions (x,y) to |p^x±q^y|=c and at most two solutions (x,y,z) to p^x±q^y±2^z=0, for given positive primes p and q and integer c.     

Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

The complete solution in integers of x3 + 3y3 = 2n

Some general remarks are made concerning the equation f(x, y) = qⁿ in the integral unknowns x, y, n, where f is an integral form and q > 1 is a given integer. It is proved that the only integral triads (x, y, n) satisfying x³ + 3y³ = 2ⁿ are (x, y, n) = (?1, 1, 1), (1, 1, 2), (?7, 5, 5,), (5, 1, 7).     

On elliptic curves y=x−nx with rank zero

In this paper we determine all elliptic curves E_n:y²=x³−n²x with the smallest 2-Selmer groups S_n=Sel₂(E_n(Q))={1} and S_n′=Sel₂(E_n′(Q))={±1,±n}(E_n′:y²=x³+4n²x) based on the 2-descent method. The values of n for such curves E_n are described in terms of graph-theory language. It is well known that the rank of the group E_n(Q) for such curves E_n is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers.     

A proof of the q,t-Catalan positivity conjecture

We present here a proof that a certain rational function C_n(q,t) which has come to be known as the “q,t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that C_n(q,t) is the Hilbert series of the diagonal harmonic alternants in the variables (x₁,x₂,…,x_n;y₁,y₂,…,y_n). Since C_n(q,t) evaluates to the Catalan number at t=q=1, it has also been an open problem to find a pair of statistics a(π),b(π) on Dyck paths π in the n×n square yielding C_n(q,t)=∑_πt^a(π)q^b(π). Our proof is based on a recursion for C_n(q,t) suggested by a pair of statistics a(π),b(π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363).     

Further examples of maximal curves

It is shown that the curve over F_q²ⁿ with n≥3 odd, that generalizes Serre’s curve y⁴+y=x³ over F₆₄, is also maximal. We also investigate a family of maximal curves over F_q²ⁿ and provide isomorphisms between these curves.     

Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 ⁿ generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n?1 primitive idempotents ofR ₂ ⁿ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n?1 idempotents are also obtained. The casen=2, 3 is dealt separately.     

On the set J(v) for Steiner Quadruple systems of order v = 2n with n⩾4

In the area of the Block-Intersection problem for Steiner Quadruple Systems (see [4, 5]), we prove that q₁₆?37 = 103 and q₁₆?29 = 111 ?J (16), and that q_v?h?J(v) for h = 21, 25, v = 2ⁿ and n?4.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

An infinite family of symmetric designs

In this paper, using the construction method of [3], we show that if q>2 is a prime power such that there exists an affine plane of order q?1, then there exists a strongly divisible 2?(q?1)(q^h?1), q^h?1(q?1), q^h?1) design for every h?2. We show that these quasi-residual designs are embeddable, and hence establish the existence of an infinite family of symmetric 2?(q^h+1?q+1,q^h, q^h?1) designs. This construction may be regarded as a generalisation of the construction of [1, Chapter 4, Section 1] and [4].     

The exponential Lebesgue-Nagell equation X 2 + P 2m = Y n

Let p be an odd prime. In this paper, a complete classification of all positive integer solutions (x, y, m, n) of the equation x ²+p ^2m = y ⁿ , gcd(x, y) = 1, n > 2, is given. As a consequence, we solve the equation for certain interesting cases.     

On large maximal partial ovoids of the parabolic quadric Q(4, q)

We use the representation ${T_2(\mathcal{O})}$ for Q(4, q) to show that maximal partial ovoids of Q(4, q) of size q ² ? 1, q = p ^h , p an odd prime, h > 1, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for q an odd prime.     

Translation planes constructed by multiple hyper-regulus replacement

New classes of mutually disjoint hyper-reguli of order q ⁿ , for n > 2, are determined, which are not André hyper-reguli if n > 3. If n is odd, each hyper-regulus permits at least two replacements and if q is odd and (n, q?1) = 1, there are (q?1)/2 mutually disjoint hyper-reguli each of which may be replaced at least two ways. Each of the possible 2^(q-1)/2 translation planes constructed is not André or generalized André. There are also new constructions of mixed subgeometry partitions.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n ² ? 1 and 12n ² + 1 are odd primes, then the general elliptic curve y ² = x ³+(36n ²?9)x?2(36n ²?5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y ² = x ³ + 27x ? 62 really is an unusual elliptic curve which has large integral points.     

The embedding of (0, α)-geometries in PG(n, q): part II

In Part I we obtained results about the embedding of (0, α)-geometries in PG(3, q). Here we determine all (0, α)-geometries with q+1 points on a line, which are embedded in PG(n, q), n>3 and q>2. As a particular case all semi partial geometries with parameters s=q,t,α(>1),μ, which are embeddable in PG(n, q), q≠2, are obtained. We also prove some theorems about the embedding of (0, 2)-geometries in PG(n, 2): we show that without loss of generality we may restrict ourselves to reduced (0, 2)-geometries, we determine all (0, 2)-geometries in PG(4, 2), and we describe an unusual embedding of U_2,3(9) in PG(5, 2).     

New relationships between steenrod operations for certain spaces

In this paper, we show that under a certain technical condition, if a space has no 2-torsion, then eitherS q ² n x≠0 or there exists somey withS q ² n y=S q ² n ⁺¹ x, if for somen≥3S q ² n ⁺¹ x≠0. The proof uses relations between Steenrod operations and operations in connective realK-Theory. This research was partially supported by NSERC.     

A new construction of caps

In this paper we present a general method to construct caps in higher-dimensional projective spaces. As an application, for q≥8 even we obtain caps in PG(5,q) larger than the caps known so far, and a new class of caps of size (q+1)(q²+3) for q≥7 odd.     

Lower bounds for the cardinality of minimal blocking sets in projective spaces

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r₂(q) be the number such that q+r₂(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most q^m+q^m−1+…+q+1+r₂(q)·(∑_i=2m−n′^m−1qⁱ) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals q^m+q^m−1+…+q+1+r₂(q)(∑_i=2m−n′^m−1qⁱ). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<q^m+q^m−1+…+q+1+r₂(q)(q^m−1+q^m−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. For q=p^3h, p7 and q not a square we show this assertion for |B|q^m+q^m−1+…+q+1+q^2/3·(q^m−1+…+1).