Related papers: Bounding the signed count of real bitangents to pl…

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic…

We generalize a previous result by Fabricius-Bjerre from curves in $\mathbb R^2$ to curves in $\mathbb R P^2$. Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson-Vogt…

Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on a smooth cubic surface over arbitrary fields. Their approach using $\mathbb{A}^1$-enumerative geometry suggests that other classical enumerative problems should…

Aronhold's classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e. special 7-tuples of bitangents such that the six points at which any subtriple of bitangents touches…

A plane quartic curve is called L\"uroth if it contains the ten vertices of a complete pentalateral. White and Miller constructed in 1909 a covariant quartic 4-fold, associated to any plane quartic. We review their construction and we show…

We study the sets of planes in an even dimensional real vector space $V$ which are simultaneously stabilised by a pair of complex structures on $V$. We completely describe these sets of planes for pairs of orthogonal complex structures.…

Every polygon with n vertices in the complex projective plane is naturally associated with its adjoint curve of degree n-3. Hence the adjoint of a heptagon is a plane quartic. We prove that a general plane quartic is the adjoint of exactly…

We prove that a natural signed count of the $2$-torsion points on a real principally polarized abelian variety $A$ always equals to $2^{g}$ where $g$ is the dimension of $A$. When $A$ is the Jacobian of a real curve we derive signed counts…

We resolve a conjecture of Albert and Bousquet-Melou enumerating quarter-plane walks with fixed horizontal and vertical projections according to their upper-right-corner count modulo 2. In doing this, we introduce a signed…

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…

We prove that every finite family of convex sets in the plane satisfying the $(4,3)$-property can be pierced by $9$ points. This improves the bound of $13$ proved by Gy\'arf\'as, Kleitman, and T\'oth in 2001.

In this paper, we enumerate with signs the holomorphic maps between Real Riemann surfaces. We relate the signs and the numbers obtained to the Real Gromov-Witten theory of the target. For a finite group $G$ with a non-trivial morphism to…

We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either…

We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can be used to show some enumerative…

We show that the signed counts of normalized real polynomials, as defined by Itenberg and Zvonkine, provide the signed counts of genus zero real ramified coverings of the Riemann sphere with a point of total ramification and several other…

We show that for all real biquadratic fields not containing $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7},$ and $\sqrt{13}$, the Pythagoras number of the ring of algebraic integers is at least $6$. We will also provide an upper…

Suppose that a polygon $P$ is given as an array containing the vertices in counterclockwise order. We analyze how many vertices (including the index of each of these vertices) we need to know before we can bound $P$, i.e., report a bounded…

In the paper we prove that the number of graphs inscribed into graph of a convex polyhedron and circumscribed around another graph does not exceed 4. For this we first studied Poncelet type problem about the number of convex $n$-gons…

We give a sharp lower bound to the largest possible Euclidean norm of signed sums of $n$ vectors in the plane. This is achieved by connecting the signed vector sum problem to the isoperimetric problem for the circumradius of polygons. In…

We transform the positive-genus real Gromov-Witten invariants of many real-orientable symplectic threefolds into signed counts of curves. These integer invariants provide lower bounds for counts of real curves of a given genus that pass…