Viro’s method and T-curves.

*(English)*Zbl 0879.14029
González-Vega, Laureano (ed.) et al., Algorithms in algebraic geometry and applications. Proceedings of the MEGA-94 conference, Santander, Spain, April 5-9, 1994. Basel: Birkhäuser. Prog. Math. 143, 177-192 (1996).

The paper presents some results obtained by means of the Viro method to construct plane real algebraic curves of a given degree with a prescribed oval arrangement. The construction discussed is of purely combinatorial nature: given a convex lattice polygon, its (lattice) triangulation, and a distribution of signs at the vertices of the triangulation, there can be constructed a broken line in the plane which is isotopic to a real non-singular curve defined by a polynomial with the given lattice polygon as its Newton polygon.

As application the author refers to his earlier construction of counterexamples to Ragsdale’s conjecture [I. Itenberg, C. R. Acad. Sci., Paris, Sér. I 317, No. 3, 277-282 (1993; Zbl 0787.14040)], and gives the complete classification of M-curves of degree \(4k+2\) with one non-empty oval: for any positive integer \(p,n\) satisfying \(p+n=2k(4k+1)+1\) (the maximal possible number of ovals for the degree \(4k+2\)), \(p-n\equiv 1\) mod 8 (Gudkov-Rokhlin congruence), \(\max\{p,n\}\leq 6k^2+3k+1\) (Petrovsky inequalities), there exists a real curve of degree \(4k+2\) with one non-empty oval, \(n\) empty ovals inside the non-empty one, and \(p-1\) empty ovals outside.

For the entire collection see [Zbl 0841.00016].

As application the author refers to his earlier construction of counterexamples to Ragsdale’s conjecture [I. Itenberg, C. R. Acad. Sci., Paris, Sér. I 317, No. 3, 277-282 (1993; Zbl 0787.14040)], and gives the complete classification of M-curves of degree \(4k+2\) with one non-empty oval: for any positive integer \(p,n\) satisfying \(p+n=2k(4k+1)+1\) (the maximal possible number of ovals for the degree \(4k+2\)), \(p-n\equiv 1\) mod 8 (Gudkov-Rokhlin congruence), \(\max\{p,n\}\leq 6k^2+3k+1\) (Petrovsky inequalities), there exists a real curve of degree \(4k+2\) with one non-empty oval, \(n\) empty ovals inside the non-empty one, and \(p-1\) empty ovals outside.

For the entire collection see [Zbl 0841.00016].

Reviewer: E.I.Shustin (Tel Aviv)

##### MSC:

14P25 | Topology of real algebraic varieties |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

14Q05 | Computational aspects of algebraic curves |