A conic and an -quintic with a point at infinity.
A group law on smooth real quartics having at least real branches
Let be a smooth real quartic curve in . Suppose that has at least real branches . Let and let . Let be the map from into the neutral component Jac of the set of real points of the jacobian of , defined by letting be the divisor class of the divisor . Then, is a bijection. We show that this allows an explicit geometric description of the group law on Jac. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of...
A local Bernstein inequality on real algebraic varieties.
A really elementary proof of real Lüroth's theorem.
Classical Lüroth theorem states that every subfield K of K(t), where t is a transcendental element over K, such that K strictly contains K, must be K = K(h(t)), for some non constant element h(t) in K(t). Therefore, K is K-isomorphic to K(t). This result can be proved with elementary algebraic techniques, and therefore it is usually included in basic courses on field theory or algebraic curves. In this paper we study the validity of this result under weaker assumptions: namely, if K is a subfield...
A set on which the Łojasiewicz exponent at infinity is attained
We show that for a polynomial mapping the Łojasiewicz exponent of F is attained on the set .
About the image of the total signature map in the two-dimensional case.
Algebraic, Rational, and homological equivalence of real cycles.
Algebraically constructible functions and signs of polynomials.
Algebraicity of the ample cone of projective varieties.
All compact manifolds are homeomorphic to totally algebraic real algebraic sets.
An algebraic method for calculating the topological degree
Approximation by continuous rational maps into spheres
Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps.
Behavior of Welschinger invariants under Morse simplifications
Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves
We show that in the class of compact, piecewise curves K in , the semialgebraic curves are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for the derivatives of (the traces of) polynomials on K.
Bounding the Number of Connected Components of a Real Algebraic Set.
Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities
We show that in the class of compact sets K in with an analytic parametrization of order m, the sets with Zariski dimension m are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on K.
Characterization of smooth, compact algebraic curves in ℝ²
Compactifications of moduli spaces in real algebraic geometry.
Computation of Period Matrices of Real Algebraic Curves*
Construction d'hypersurfaces réelles