Digital methodology for analyzing the yield of sugar beet on leached chernozem of the Krasnodar Territory

In the article, a mathematical expression is obtained that characterizes the effect of the doses of introduced fertilizers, the content of nutrients in the soil and their absorption by sugar beet plants on the yield of this crop. With the help of the developed mathematical model, the optimal values of the content of macronutrients in soil and plants are determined: nitrogen ~ 21.81 mg/kg and 1.02 %; phosphorus ~ 58.88 mg/kg and 0.33 %; potassium ~ 120.21 mg/kg and 1.33%, respectively, which accumulate when introducing a complete fertilizer at a dose of N80P80K80. This leads to the achievement of the maximum yield of root crops, which corresponds to the calculated value of this indicator ≈of 56.94 and 64.6 t/ha and coincides with the experimental one = 57.9 and 62.0 t/ha, respectively, obtained in this variant. The introduced fertilizers containing boron contributed to a more intensive absorption of nitrogen, phosphorus and potassium by root crops. The use of trace elements causes a tendency to increase the yield of this crop.


Introduction
For the Russian Federation, sugar beet is the main raw material for sugar production. Over the past three years, the area of this crop sowings in the Russian Federation amounted to 1063.2 thousand. ha, including 191.8 thousand hectares in the Krasnodar Territory. The yield of root crops in the country on average was 380.4 c/ha (gross harvest of 48.0 million tons), in the Krasnodar Territory 522.4 c/ha, (gross harvest of 8.0 million tons) [3,4 ].
Fertilizers are an essential factor in stabilizing the yield of many agricultural crops, sugar beet is no exception, and their influence on the formation of this crop productivity is 26,8 % [1,3,9].

Materials and methods
The analysis of experimental data was carried out on the basis of studies conducted in the stationary field experiment of the Department of Agrochemistry of the Kuban State

Results and discussion
The introduced fertilizers had a positive effect on the studied indicators for the development of sugar beet, which led to an increase in the number of forms of nitrogen, phosphorus and potassium available for plant nutrition in the soil [4,9].
Thus, in the phase of industrial ripeness of root crops, the level of mineral nitrogen on the fertilized variants remained on average 7.0 mg/kg higher than on the non-fertilized variant (Table 1). Its quantity has especially increased in the variants with unilateral nitrogen application (N40P0K0) and complete fertilizer (N80P80K80). The data in Table 1 show that the content of mobile phosphorus increased on the fertilized variants by an average of 17.2 mg/kg compared to the control one. The best variants were N0P80K0 and N80P80K80. The amount of available potassium in the soil was higher in the variants with the use of fertilizers by 23.8 mg/kg. The highest residue of this element was in the variants N0P0K80 and N120P120K120.
An increase in the level of soil nutrition of sugar beet due to the use of mineral fertilizers led to an increase in the absorption of nutrients by plants ( Table 2). The data in Table 2 show that the amount of nitrogen in sugar beet root crops increased by an average of 0.1 %, phosphorus by 0.04% and potassium by 0.13% due to the use of fertilizers. Very effective results were obtained with a balanced mineral nutrition of plants in the variant N80P80K80.
Experiments have shown that the yield of sugar beet root crops has increased due to the use of fertilizers (Table 3). The data in Table 3 show that the yield of root crops on fertilized variants increased by 11.9 t/ha or 26.4% on average according to the experiment. The greatest increase in yield was obtained on the variants using N80P80K80 and N120P120K120, but the best sugar content was when using N80P80K80 (18.7%), which indicates that it is not advisable to increase the dose of fertilizers to a high level.
Sugar beet plants during the growing season need not only nitrogen, phosphorus and potassium, but also trace elements, mostly boron, for the normal course of physiological and biochemical processes [3].
Experiments have shown that the treatment of sugar beet plants with fertilizer solutions with trace elements provided a more intensive absorption of macronutrients (Table 4). The results showed that the accumulation of nitrogen in root crops increased by an average of 0.15 %, an increase from 0.09 to 0.19 % relative to the background. The phosphorus content ranged from 0.29-0.34 % and increased by 0.03% on average. The potassium content increased by 1.20-1.32 % and on average exceeded the background by 0.13 %. Boric micro-fertilizer had the maximum effect on all indicators.
The applied trace elements, increasing the absorption of nitrogen, phosphorus and potassium by plants, contributed to the formation of a greater yield of root crops (Table 5). Under the influence of micro-fertilizers, there is a tendency to increase the amount of crop. On average, this indicator increased by 1.5 t/ha. Mathematical processing. In contrast to the most popular methods of processing of research results (regression) with a single factor [5], a mathematical model of multiple nonlinear correlation was used for the data under consideration [2,6]. In [6], it is proposed to build mathematical models for the production function of the efficiency of fertilizer use in the form of polynomials of the following type: It is also noted in [6] that "the processing of actual experimental data to determine the numerical values of parameters 0 1 9 , , , a a a … , the sum of regression squares and the residual sum of squares for each of these models is carried out, as usual, by the least squares method.
Calculations of equations with 10 components, especially for fractional powers of variables, are lengthy and require the use of a computer." The following functional dependence is taken as a mathematical model that determines the yield of sugar beet: (1) According to the ratio (1), the yield (Ni) of sugar beet is determined by the amount of nitrogen (xi), phosphorus (yi) and potassium (zi) introduced in the form of fertilizers and the content of macronutrients in the soil in the amount of mg per 1 kg of soil; 0 1 2 3 , , , , а а а с … . -constant coefficients determined by the least squares' method based on experimental data.
The values of the ten constant coefficients 0 1 2 3 1 2 3 , , , , , , , а а а а b b b 1 2 3 , , c c c are found from the condition of the minimum of the sum of the squares of deviations of experimental data of P0(xi, yi, zi) from the calculated values Pof p(xi, yi, zi), determined by the formula (1) and, therefore, we can write the ratio: (2) n -the number of experiments (trials)i ∆ -the sum of the squares of the deviations of the experimental data P0 (xi, yi, zi) from the calculated values Pp (xi, yi, zi).
Thus, the values of these unknown constants are found from a system of ten algebraic linear equations of the form: (3) (the summation index (i) is omitted for brevity).
In the expanded form, the equations of the system (3) have the form: To solve the considered system (4), any known method can be used, for example, the Gauss or Kramer method.
After the values of the constant coefficients 0 1 2 3 1 2 3 1 2 3 , , , , , , , , , а а а а b b b c c c are determined, the function of the three arguments P(x, y, z) of the relation (1) allows to find the "critical" (optimal) values of the parameters xcr, ycr, zcr from the system of three algebraic equations: (5) or, in the expanded form: (6) To reduce the amount of computational work and improve the accuracy of calculations, it is advisable to convert the experimental data to dimensionless units: the values of the first row are taken as a conditional unit, and the subsequent values are determined as a quotient of the dimensional data to the values of the first row.
Based on the obtained dimensionless data, the values of the coefficients of the system of ten linear algebraic equations (4) were determined, which can be written as:  , , , , , , , , , а а а а b b b c c c , it is possible to create a system of three linear algebraic equations (6) to determine the "optimal" values of macronutrients: xcr, ycr, zcr in the form: (10) The solution of this system is the values: (11) For these values of the arguments, we find the highest yield value in dimensionless units: Pp (xcr, ycr, zcr) = 1,646 In dimensional units, you can find the values: and, therefore, the maximum yield is equal to: potassium -1.33%, at which the estimated maximum yield is 64.6 t/ha, corresponding to an experiment average value of 62.0 t/ha, which is obtained in the variant with the introduction of N80P80K80. 4. The effect of fertilizing plants with boric micro-fertilization led to an increase in the absorption of nutrients by sugar beet plants, which corresponds to the content of macronutrients in root crops: nitrogen -1.49%; phosphorus -0.37%, potassium -1.49 %, with a yield of 62.9 t/ha, which was obtained in the background variant.