Serdica Mathematical Journal

Volume 31, Number 3, 2005

C O N T E N T S

• Ballico, E., C. Keem. Projectively normal line bundles on k-gonal curves and rational surfaces (pp. 175-188)
• Ghéribi-Aoulmi, Z., M. Bousseboua. Recursive methods for construction of balanced n-ary block designs (pp. 189-200)
• Kostov, Vl. P. On root arrangements of polynomial-like functions and their derivatives (pp. 201-216)
• Michailov, I. Quaternion extensions of order 16 (pp. 217-228)
• Sarti, A. A geometrical construction for the polynomial invariants of some reflection groups (pp. 229-242)
• Stoimenova, V. Robust parametric estimation of Branching processes with a Random Number of Ancestors (pp. 243-262)

A B S T R A C T S

PROJECTIVELY NORMAL LINE BUNDLES ON k-GONAL CURVES AND RATIONAL SURFACES
E. Ballico ballico@science.unitn.it,
C. Keem ckeem@math.snu.ac.kr

2000 Mathematics Subject Classification: 14H50.
Key words: plane curve, singular curve, gonality, scrollar invariants.

Here we prove the projective normality of several special line bundles on a general k-gonal curve. Let X be a k-gonal curve arising as the normalization of a certain nodal curve Y Ì P¹× P¹. We prove the existence of many projectively normal special line bundles on X. We also show the existence of a large set, F, of special line bundles on X which are not projectively normal (and not even quadratically normal) and for every L Î F we compute the dimension of the cokernel of the multiplication map H⁰(X,L)ÄH⁰(X,L)® H⁰(X,L^Ä2). Let M be the blowing - up either of P² or of P¹×P¹ at a general finite set S. We show the projective normality of certain line bundles on M, the case P¹× P¹ being used to prove our results on k-gonal curves.

RECURSIVE METHODS FOR CONSTRUCTION OF BALANCED n-ARY BLOCK DESIGNS
Z. Ghéribi-Aoulmi gheribiz@yahoo.fr,
M. Bousseboua

2000 Mathematics Subject Classification: Primary 05B05; secondary 62K10.
Key words: Balanced incomplete binary blocks, n-ary designs, finite projective geometry, finite linear sub-variety.

This paper presents a recursive method for the construction of balanced n-ary block designs.

This method is based on the analogy between a balanced incomplete binary block design (B.I.E.B) and the set of all distinct linear sub-varieties of the same dimension extracted from a finite projective geometry. If V₁ is the first B.I.E.B resulting from this projective geometry, then by regarding any block of V₁ as a projective geometry, we obtain another system of B.I.E.B. Then, by reproducing this operation a finite number of times, we get a family of blocks made up of all obtained B.I.E.B blocks. The family being partially ordered, we can obtain an n-ary design in which the blocks are consisted by the juxtaposition of all binary blocks completely nested. These n-ary designs are balanced and have well defined parameters. Moreover, a particular balanced n-ary class is deduced with an appreciable reduction of the number of blocks.

ON ROOT ARRANGEMENTS OF POLYNOMIAL-LIKE FUNCTIONS AND THEIR DERIVATIVES
Vladimir Petrov Kostov kostov@math.unice.fr

2000 Mathematics Subject Classification: 12D10.
Key words: hyperbolic polynomial; root arrangement; configuration vector.

A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. Denote by x_k⁽ⁱ⁾ the roots of P⁽ⁱ⁾, k = 1,¼,n-i, i = 0,¼,n-1. Then one has "i < j, x_k⁽ⁱ⁾ £ x_k^(j) £ x_k+j-i⁽ⁱ⁾ and ((x_k⁽ⁱ⁾ = x_k⁽ⁱ⁺¹⁾) or (x_k+1⁽ⁱ⁾ = x_k⁽ⁱ⁺¹⁾))Þ (x_k⁽ⁱ⁾ = x_k⁽ⁱ⁺¹⁾ = x_k+1⁽ⁱ⁾). For n ³ 4 not all arrangements of n(n+1)/2 real numbers x_k⁽ⁱ⁾ compatible with these two conditions are realizable by the roots of hyperbolic polynomials of degree n and of their derivatives. We show that for n = 4 they are realizable either by hyperbolic polynomials of degree 4 or by non-hyperbolic polynomials of degree 6 whose fourth derivatives never vanish (these are a particular case of the so-called hyperbolic polynomial-like functions of degree 4).

QUATERNION EXTENSIONS OF ORDER 16
Ivo M. Michailov ivo_michailov@yahoo.com

2000 Mathematics Subject Classification: 12F12.
Key words: embedding problem, extension, Galois, quaternion.

We describe several types of Galois extensions having as Galois group the quaternion group Q₁₆ of order 16.

A GEOMETRICAL CONSTRUCTION FOR THE POLYNOMIAL INVARIANTS OF SOME REFLECTION GROUPS
Alessandra Sarti sarti@mathematik.uni-mainz.de

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.
Key words: Polynomial invariants, Reflection and Coxeter groups, Group actions on varieties.

We construct invariant polynomials for the reflection groups [3, 4, 3] and [3, 3, 5] by using some special sets of lines on the quadric P₁×P₁ in P₃. Then we give a simple proof of the well known fact that the ring of invariants are rationally generated in degree 2, 6, 8, 12 and 2, 12, 20, 30.

ROBUST PARAMETRIC ESTIMATION OF BRANCHING PROCESSES WITH A RANDOM NUMBER OF ANCESTORS
Vessela Stoimenova stoimenova@fmi.uni-sofia.bg

2000 Mathematics Subject Classification: 60J80.
Key words: Branching processes, random number of ancestors, power series distribution, parametric estimation, robustness, d-fullness.

The paper deals with a robust parametric estimation in branching processes {Z_t(n)} having a random number of ancestors Z₀(n) as both n and t tend to infinity (and thus Z₀(n) in some sense). The offspring distribution is considered to belong to a discrete analogue of the exponential family - the class of the power series offspring distributions. Robust estimators, based on one and several sample paths, are proposed and studied for all values of the offspring mean m, 0 < m < ¥, in the subcritical, critical and supercritical case.