Real Algebraic Curves Research Articles

Real inflection points of real hyperelliptic curves

Given a real hyperelliptic algebraic curve $X$ with non-empty real part and a real effective divisor $\mc{D}$ arising via pullback from $\mathbb{P}^1$ under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|\mc{D}|$ on $X$. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces.

Quasi-Nash varieties and Schwartz functions on them

We introduce a new category called Quasi-Nash, unifying Nash manifolds and algebraic varieties. We define Schwartz functions, tempered functions and tempered distributions in this category. We show that the classical properties of these spaces, that hold on Nash manifolds and real affine algebraic varieties, hold in this category as well.

Approximation of maps into spheres by piecewise-regular maps of class C^k

The aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise-regular maps of class {mathcal {C}}^k, where k is an arbitrary nonnegative integer.

Real algebraic curves with large finite number of real points

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree. Some of the results are extended to other real algebraic surfaces, most notably ruled.

A Hybrid Procedure for Finding Real Points on a Real Algebraic Set

Motivated by the idea of Shen, et al.’s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis, this paper presents a hybrid procedure to compute sample points on each connected component of a real algebraic set by combining a special homotopy method and interval analysis with a better estimate on initial intervals. For a real algebraic set given by a polynomial system, the new method first constructs a square polynomial system which represents the sample points, and then solve this system by a special homotopy continuation method introduced recently by Wang, et al. (2017). For each root returned by the homotopy continuation method, which is a complex approximation of some (complex/real) root of the polynomial system, interval analysis is used to verify whether it is an approximation of a real root and finally get real points on the given real algebraic set. A new estimate on initial intervals is presented which helps compute smaller initial intervals before performing interval iteration and thus saves computation. Experiments show that the new method works pretty well on tested examples.

Global Optimization of Polynomials over Real Algebraic Sets

Let f, g1,..., gs be polynomials in R[X1,..., Xn]. Based on topological properties of generalized critical values, the authors propose a method to compute the global infimum f* of f over an arbitrary given real algebraic set V = {x ∈ Rn | g1(x) = 0,..., gs(x) = 0}, where V is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing f over a basic closed semi-algebraic set S = {x ∈ Rn | g1(x) ≥ 0,..., gs(x) ≥ 0}.

Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

A Gradient Sampling Method on Algebraic Varieties and Application to Nonsmooth Low-Rank Optimization

In this paper, a nonsmooth optimization method for locally Lipschitz functions on real algebraic varieties is developed. To this end, the set-valued map $\varepsilon$-conditional subdifferential $x\to \partial_{\varepsilon}^{N} f(x):= \partial_{\varepsilon}f(x)+N(x)$ is introduced, where $\partial_{\varepsilon}f(x)$ is the Goldstein-$\varepsilon$-subdifferential and $N(x)$ is a closed convex cone at $x$. It is proved that negative of the shortest $\varepsilon$-conditional subgradient provides a descent direction in $T(x)$, which denotes the polar of $N(x)$. The $\varepsilon$-conditional subdifferential at an iterate $x_{\ell}$ can be approximated by a convex hull of a finite set of projected gradients at sampling points in $x_\ell+\varepsilon_{\ell} B_{T(x_{\ell})}(0,1)$ to $T(x_{\ell})$, where $T(x_{\ell})$ is a linear space in the Bouligand tangent cone and $ B_{T(x_{\ell})}(0,1)$ denotes the unit ball in $T(x_{\ell})$. The negative of the shortest vector in this convex hull is shown to be a descent direction in the Bouligand tangent cone at $x_{\ell}$. The proposed algorithm makes a step along this descent direction with a certain step-size rule, followed by a retraction to lift back to points on the algebraic variety $\mathcal{M}$. The convergence of the resulting algorithm to a critical point is proved. For numerical illustration, the considered method is applied to some nonsmooth problems on varieties of low-rank matrices $\mathcal{M}_{\leq r}$ of real $M\times N$ matrices of rank at most $r$, specifically robust low-rank matrix approximation and recovery in the presence of outliers.

On the Geometry of the Set of Symmetric Matrices with Repeated Eigenvalues

We investigate some geometric properties of the real algebraic variety Delta of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart–Young–Mirsky-type theorem for the distance function from a generic matrix to points in Delta . We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of Delta ) and random matrix theory.

Nonexistence of torically maximal hypersurfaces

Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gau{\ss} map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.

Real Gromov-Witten theory in all genera and real enumerative geometry: Construction

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for counts of real positive-genus curves in real algebraic varieties. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces; the previous attempts involved direct computations for the determinant lines of Fredholm operators over bordered surfaces. We use the notion of real orientation introduced in this paper to obtain isomorphisms of real bundle pairs over families of symmetric surfaces and then apply the determinant functor to these isomorphisms. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C.Liu's thesis for a fully fledged real Gromov-Witten theory. The second and third parts of this work concern applications: they describe important properties of our orientations on the moduli spaces, establish some connections with real enumerative geometry, provide the relevant equivariant localization data for projective spaces, and obtain vanishing results in the spirit of Walcher's predictions.

Schwartz Functions on Real Algebraic Varieties

AbstractWe define Schwartz functions, tempered functions, and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.

Algebraic stratified general position and transversality

The method of Whitney interpolation is used to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for semialgebraic subsets.

On the polynomial Wolff axioms

We confirm a conjecture of Guth concerning the maximal number of $${\delta}$$ -tubes, with $${\delta}$$ -separated directions, contained in the $${\delta}$$ -neighborhood of a real algebraic variety. Modulo a factor of $${\delta^{-{\varepsilon}}}$$ , we also prove Guth and Zahl’s generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski’s projection theorem and Gromov’s algebraic lemma.

Learning algebraic varieties from samples

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.

Compactness criteria for real algebraic sets and Newton polyhedra

Abstract Let f : ℝ n → ℝ {f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}} be a polynomial and 𝒵 ⁢ ( f ) {\mathcal{Z}(f)} its zero set. In this paper, in terms of the so-called Newton polyhedron of f, we present a necessary criterion and a sufficient condition for the compactness of 𝒵 ⁢ ( f ) {\mathcal{Z}(f)} . From this we derive necessary and sufficient criteria for the stable compactness of 𝒵 ⁢ ( f ) {\mathcal{Z}(f)} .

Irreducibility of lemniscates

We prove that lemniscates (i.e., sets of the form $|P(z)|=1$ where $P$ is a complex polynomial) are irreducible real algebraic curves.

Maximal ideals of regulous functions are not finitely generated

We prove that every maximal ideal in the ring of k-regulous functions, k∈N, on a smooth real affine algebraic variety of dimension d≥2 is not finitely generated.

On Coxeter algebraic varieties

Connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group are investigated. In particular, we consider their maximal number and their geometric and topological properties. This provides a decomposition for the space of $CB_{n}$-algebraic varieties. We construct $CB_{n}$-polynomials using Young-posets and partitions of integers. Our results establish bounds on the number of connected components for a given set of coefficients. It turns out that this number can achieve an upper bound of $2^{n}+1$ for specific coefficients. We introduce a new method to characterize the geometry of these real algebraic varieties, using J. Cerf and A. Douady theory for varieties with angular boundary and the theory of chambers and galleries. We provide several examples that bring out the essence of these results.

On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities

Abstract We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant d. We prove that if a Specht module, $\mathbb{S}^{\lambda }$, appears with positive multiplicity in the isotypic decomposition of the cohomology modules of such sets, then the rank of the partition $\lambda$ is bounded by O(d). This implies a polynomial (in the dimension of the ambient space) bound on the number of such modules. Furthermore, we prove a polynomial bound on the multiplicities of those that do appear with positive multiplicity in the isotypic decomposition of the abovementioned cohomology modules. We give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets, improving in certain situations prior results of Gabrielov, Vorobjov, and Zell.