Intersection Of Hypersurfaces Research Articles

Schmidt rank and singularities

UDC 512.5 We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous polynomials, assuming the characteristic does not divide the degree. Further, we use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces). This gives an effective version of Ananyan--Hochster's theorem [J. Amer. Math. Soc., <strong>33</strong>, No. 1, 291–309 (2020), Theorem A].

Intersections of Hypersurfaces and Ring of Conditions of a Spherical Homogeneous Space

We prove a version of the BKK theorem for the ring of conditions of a spherical homogeneous space $G/H$. We also introduce the notion of ring of complete intersections, firstly for a spherical homogeneous space and secondly for an arbitrary variety. Similarly to the ring of conditions of the torus, the ring of complete intersections of $G/H$ admits a description in terms of volumes of polytopes.

Efficient Multidimensional Diracs Estimation With Linear Sample Complexity

Estimating Diracs in continuous two or higher dimensions is a fundamental problem in imaging. Previous approaches extended one-dimensional (1-D) methods, like the ones based on finite rate of innovation (FRI) sampling, in a separable manner, e.g., along the horizontal and vertical dimensions separately in 2-D. The separate estimation leads to a sample complexity of $\mathcal {O}\left(K^D\right)$ for $K$ Diracs in $D$ dimensions, despite that the total degrees of freedom only increase linearly with respect to $D$ . We propose a new method that enforces the continuous-domain sparsity constraints simultaneously along all dimensions, leading to a reconstruction algorithm with linear sample complexity $\mathcal {O}(K)$ , or a gain of $\mathcal {O}\left(K^{D-1}\right)$ over previous FRI-based methods. The multi-dimensional Dirac locations are subsequently determined by the intersections of hypersurfaces (e.g., curves in 2-D), which can be computed algebraically from the common roots of polynomials. We first demonstrate the performance of the new multidimensional algorithm on simulated data: multidimensional Dirac location retrieval under noisy measurements. Then, we show results on real data: radio astronomy point source reconstruction (from LOFAR telescope measurements) and the direction of arrival estimation of acoustic signals (using Pyramic microphone arrays).

Laplace and Z transforms of linear dynamical systems and conic sections

We consider the solution trajectories of linear continuous and discrete dynamical systems and show that in the Laplace and Z transform spaces, respectively, they lie on the intersections of hypersurfaces described by second-degree polynomial equations. In particular, in two dimensions, these intersections are conic sections whose types are determined by the eigenvalues of the coefficient matrices.

On the Variety of Paths on Complete Intersections in Grassmannians

In this article we study the Fano variety of lines on the complete intersection of the grassmannian G(n, 2n) with hypersurfaces of degrees d 1 ..., d i . A length l path on such a variety is a connected curve composed of l lines. The main result of this article states that the space of length l paths connecting any two given points on the variety is nonempty and connected if ∑ d j < n/4 . To prove this result we first show that the space of length n paths on the grassmannian G(n, 2n) that join two generic points is isomorphic to the direct product F n ×F n of spaces of full flags. After this we construct on Fn ×Fn a globally generated vector bundle E with a distinguished section s such that the zeros of s coincide with the space of length n paths that join x and y and lie in the intersection of hypersurfaces of degrees d 1 ,...,d k . Using a presentation of E as a sum of linear bundles we show that zeros of its generic and, hence, any section form a non empty connected subvariety of F n × F n . Apart from its immediate geometric interest, this result will be used in our future work on generalisation of splitting theorems for finite rank vector bundles on ind-manifolds.

On the destruction of minimal foliations

A generalisation of Waring's problem, considered first by Arkhipov and Karatsuba, is the question of representing not an integer, but a given polynomial, as a sum of powers of linear polynomials. We investigate a related problem and prove a Hasse principle for the number of identical representations of a set of given forms by homogeneous polynomials of general shape. The result leads to sizeable improvements for estimates of the number of linear spaces on the intersection of hypersurfaces.

Examples of factorial threefolds complete intersection in $${\mathbb{P}^5}$$ with many singularities

Denote by νm(d) the maximal integer for which there exists for \({d \gg 0}\) a threefold \({X\subset \mathbb{P}^5}\) complete intersection of hypersurfaces of degree respectively d and d − 1 such that X has only ordinary singularities of order m and |Sing(X)| = νm(d). We prove that, \({\nu_m(d)\ge \varphi(d)}\) where \({\varphi(d)\sim d^5}\) asymptotically. This result extends (Di Gennaro and Franco in Commun Contemp Math 10(5):745–764, 2008, Corollary 2.10).

A speciality theorem for curves in P5

Let \(C{\subset}\,{\bf P}^r\) be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that \(h^0(C,\omega_C(-t)) > 0\) , where ωC denotes the dualizing sheaf of \(C\) . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if \(C {\subset}\,{\bf P}^5\) is an integral degree d curve not contained in any surface of degree > }s{ > > } t > > u\geq 1\) , then \( e(C)\leq {\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6. \) Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, \({\frac{t}{u}}\) , \({\frac{s}{t}}\) and \({\frac{d}{s}}\) . We give also some partial results in the general case \(C\subset {\bf P}^r\) , \(r\geq 3\) .

Generalized Weierstrass kernels on the intersection of two complex hypersurfaces

On plane algebraic curves the so-called Weierstrass kernel plays the same role of the Cauchy kernel on the complex plane. A straightforward prescription to construct the Weierstrass kernel has been known for more than one century. How can it be extended to the case of more general curves obtained from the intersection of hypersurfaces in an n-dimensional complex space? This problem is solved in this work in the case n=3. As an application, the correlation functions of bosonic string theories are constructed on a canonical curve of genus four.

A symbolic approximation of periodic solutions of the Henon–Heiles system by the normal form method

This article describes the computer algebra application of the normal form method for building analytic approximations for all (including complex) local families of periodic solutions in a neighborhood of the stationary point to the Henon–Heiles system. The families of solutions are represented as truncated Fourier series in approximated frequencies and the corresponding trajectories are described by intersections of hypersurfaces which are defined by pieces of multivariate power series in phase variables of the system. A comparison numerical values created by a tabulation of the approximated solutions above with results of a numerical integration of the Henon–Heiles system displays a good agreement which is enough for a usage these approximate solutions for engineering applications. Such approximations can be useful for a phase analysis of wide class of autonomous nonlinear systems with smooth enough right sides near a stationary point. The method also provides a new convenient graphic representation for the phase portrait of such systems. There is possibility for searching polynomial integral manifolds of the system. For the given example they can be evaluated in a finite form.

Left-flat spaces and null geodesics

Equations are derived giving the null geodesics in any left-flat space as the intersection of hypersurfaces. The connection with null geodesics given in terms of the good-cut functions forH spaces is established.

Complete intersections as branched covers and the Kervaire invariant

A complete intersection is the transversal intersection of hypersurfaces in complex projective space. I f X , C CP,, is the intersection of r = m n hypersurfaces we show in §2 that, up to diffeomorphism, X,, is a regular branched cyclic cover of an intersection of r 1 hypersurfaces. This extends the well-known and useful fact that a hypersurface is a regular branched cyclic cover of complex projective space. The remainder of the paper is concerned with when the Kervaire invariant of a complete intersection X, of odd complex dimension n is defined in a certain geometric way, with comput ing this invariant, and with a connected sum decomposit ion of X,. In §5 are some results on the question when the normal bundle of an embedding f depends only on the homotopy class o f f If the hypersurfaces whose intersection is X, have degrees d 1 . . . . . dr, we call d =(d 1 . . . . . d,) the multi-degree of X. The degree of X is d=lld~.

Numerical characteristics of systems of straight lines on complete intersections

We obtain an equation for the number of straight lines on the complete intersection of hypersurfaces and find Hilbert's polynomial for the variety of straight lines of a cubic three-dimensional hypersurface.