The Parity of the Number of Irreducible Factors for Some Pentanomials

It is well known that Stickelberger-Swan theorem is very important for determining reducibility of polynomials over a binary field. Using this theorem it was determined the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on. We discuss this problem for type II pentanomials namely x^m +x^{n+2} +x^{n+1} +x^n +1 \in\ IF_2 [x]. Such pentanomials can be used for efficient implementing multiplication in finite fields of characteristic two. Based on the computation of discriminant of these pentanomials with integer coefficients, it will be characterized the parity of the number of irreducible factors over IF_2 and be established the necessary conditions for the existence of this kind of irreducible pentanomials.

Parity of the Number of Irreducible Factors for Composite Polynomials

Various results on parity of the number of irreducible factors of given polynomials over finite fields have been obtained in the recent literature. Those are mainly based on Swan’s theorem in which discriminants of polynomials over a finite field or the integral ring Z play an important role. In this paper we consider discriminants of the composition of some polynomials over finite fields. The relation between the discriminants of composed polynomial and the original ones will be established. We apply this to obtain some results concerning the parity of the number of irreducible factors for several special polynomials over finite fields.

Influence of occupation number of single particle levels on K - K charge transfer in collisions of 90 keV - Ne{^9+} on Ne

The influence of the occupation of the single particle levels on the impact parameter dependent K - K charge transfer occuring in collisions of 90 keV Ne{^9+} on Ne was studied using coupled channel calculations. The energy eigenvalues and matrixelements for the single particle levels were taken from ab initio self consistent MO-LCAO-DIRAC-FOCK-SLATER calculations with occupation numbers corresponding to the single particle amplitudes given by the coupled channel calculations.

Effects of row spacing on productivity and nodulation of two soybean varieties under hot sub-moist tropical conditions in south-western Ethiopia

The objective of this study was to determine the optimum row spacing to improve the productivity of two soybean (Glycine max L.) varieties under the tropical hot sub-moist agroecological conditions of Ethiopia. A two-year split-plot design experiment was conducted to determine the effect of variety (Awasa-95 [early-maturing], Afgat [medium-maturing]) and row spacing (RS: 20, 25, 30, 35, 40, 45, 50, 55, 60 cm) on the productivity, nodulation and weed infestation of soybean. Seed and total dry matter (TDM) yield per ha and per plant, and weed dry biomass per m^2 were significantly affected by RS. Soybean variety had a significant effect on plant density at harvest and some yield components (plant height, number of seeds/pod, and 1000 seed weight). Generally, seed and TDM yield per ha and per plant were high at 40 cm RS, and weed dry biomass per m^2 was higher for RS >= 40 cm than for narrower RS. However, the results did not demonstrate a consistent pattern along the RS gradient. The medium-maturing variety Afgat experienced higher mortality and ended up with lower final plant density at harvest, but higher plant height, number of seeds per pod and 1000 seed weight than the early-maturing variety Awasa-95. The results indicate that 40 cm RS with 5 cm plant spacing within a row can be used for high productivity and low weed infestation of both soybean varieties in the hot sub-moist tropical environment of south-western Ethiopia.