A note on the determinant 308 in Proskuryakov's linear algebra book

We put in evidence and correct a mistake in the formula for the determinant 308 in Proskuryakov’s linear algebra book. We apply this formula to reprove the well-known fact that the Fubini-Study metric on the complex projective space is Einstein

On the intrinsic complexity of the arithmetic Nullstellensatz

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.Facultad de Ciencias Exacta

On the intrinsic complexity of the arithmetic Nullstellensatz

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.Facultad de Ciencias Exacta

On the intrinsic complexity of the arithmetic Nullstellensatz

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function πS(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.Facultad de Ciencias Exacta

Biomass flow in massai grass fertilized with nitrogen under intermittent stocking grazing with sheep.

Abstract: This study evaluated the biomass flow of massai grass in regrowth subjected to different nitrogen levels (control - no fertilization; 400; 800; and 1200 N kg ha-1year-1) and under rotational stocking with sheep, in a completely randomized design with repeated measures over time. The leaf elongation rate increased with increasing nitrogen levels (N) and the fourth grazing cycle presented a lower value compared with the others. The stem elongation rate responded linearly with increasing levels of N, but it was not influenced by grazing cycles. The senescence rate of leaves remaining before grazing and after grazing were not influenced by the nitrogen fertilization or amended with successive grazing cycles. The phyllochron was influenced only by the nitrogen fertilization, with a decreasing linear response with nitrogen levels. Quadratic response was observed for the average length of leaves with increasing N levels and the grazing cycle 4 presented the lowest value for this variable. For each kilogram N ha-1 year-1, increases of 0.161 and 0.1604 kg ha-1 day-1 were verified in the herbage growth and accumulation rates, respectively. Nitrogen fertilization favors the biomass flow of massai grass, promoting an expressive increase in the production and accumulation of forage

Dinâmica de perfilhamento em capim-massai adubado com nitrogênio e pastejado por ovinos.

Resumo: Avaliou-se a dinâmica de perfilhamento do capim-massai sob lotação rotativa com ovinos e adubado com nitrogênio (controle - 0; 400; 800 e 1200 kg N?ha-1?ano-1), num delineamento inteiramente casualizado com medidas repetidas no tempo. As taxa de aparecimento, sobrevivência e mortalidade perfilhos responderam crescentemente às doses de nitrogênio, com as duas primeiras taxas revelando ligeira variação entre os períodos de avaliação. Constatou-se resposta quadrática, alcançando valores máximos com o incremento das doses de nitrogênio para a taxa de florescimento de perfilhos, (0,051 perfilhos?100 perfilhos-1?dia-1, respectivamente) nas doses 613,5 kg de N?ha-1?ano-1, respectivamente. Para a taxa de florescimento, verificou-se oscilação entre os períodos de avaliação. A adubação nitrogenada proporciona mudanças positivas na dinâmica de perfilhamento do capim-massai. Tillering dynamics in massai grass fertilized with nitrogen and grazed by sheep. Abstract: The tillering dynamics in massai grass subjected to different nitrogen (N) fertilization levels (control - 0; 400; 800 and 1200 kg N?ha-1?year-1) and under rotational stocking with sheep was evaluated. A completely randomized design with measurements repeated in time was adopted. The tiller appearance, survival and mortality rates were increased with the nitrogen levels, with the two rate revealing slight variation between the periods of evaluation. There was showed quadratic response reaching maximum values with increasing levels of nitrogen for tillers flowering rate (0.051 tillers?100 tillers-1?day-1, respectively) for levels 613.5 kg of N?ha-1?year-1, respectively. For the flowering rate and reproductive basal tillers population density was differences observed between the evaluation periods. The nitrogen fertilization provide positive responses to the tillering dynamics of massai grass