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About mutual positions of a cubic and two conics in projective plane

Student: Puhova Alina

Supervisor: Grigory Polotovskiy

Faculty: Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod)

Educational Programme: Mathematics (Bachelor)

Year of Graduation: 2020

The research aims to reveal the M-curves of degree 7 that decay into two conics and a cubic under natural conditions. We show different types of M-curves in projective plane under initial conditions, reduce their number and classify them. The problem of a systematic study of the topology of real algebraic curves that decompose into the product of two nonsingular curves was first posed by D.A. Gudkov in 1969 for the curve of degree 6. G.M. Polotovsky (1979) has provided ample evidence for this problem under additional conditions of maximality and the general position of the curves of the factors. We use the insights of some researchers to assist in this task. In this paper we want to deepen and push forward the study of a curve of degree 7. In this paper, I will use different methods for studying M-curves of degree 7. There is a great variety of means at our disposal: 1. The Orevkov method. 2. The Bezout’s theorem. 3. The Brusotti’s theorem. 4. Murasugi-Tristram inequality. One of the methods which I use in my paper is the Bezout's theorem. . The Bezout's theorem states that the number of common points of two such curves is at most equal to the product of their degrees, and equality holds if one counts points at infinity and points with complex coordinates, and if each point is counted with its intersection multiplicity. This method was chosen because with the help of this theorem I can also prohibit some possibilities and reduce the number of pictures. After the application of the Bezout’s theorem, some of the remaining cases contradict to the classification of cubic and conic curve. Using this method, I reduce the number of possible patterns for my curves thus facilitating further work. In this paper, a new algorithm is developed for M-curves of degree 7, the main aim of which is to research these curves and keep only valid patterns. We identified different M-curves of degree 7 in projective plane under natural conditions, reduced the number of possible patterns for different curves and classified them according to the algorithm. In this paper we studied and promoted forward the study of curve of degree 7. Our results will open different opportunities for future research into M-curve of degree 7. The results achieved are promising and indicate the value of continuing research in this direction. This study can be continued if the initial conditions are changed. As a result, we can consider different curves of degree 7 that decay into two conics and a cubic.

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