Algebraic Curve Of Degree Research Articles

Polynomial differential systems with hyperbolic limit cycles

One of the main problems of the qualitative theory of the planar differential equations is the study their limit cycles. In this paper given an algebraic curve of degree n we characterize the planar polynomial differential systems of degree greater or equal than n which admit the ovals of the algebraic curve as hyperbolic limit cycles.

Biased 2 times 2 periodic Aztec diamond and an elliptic curve

We study random domino tilings of the Aztec diamond with a biased 2 times 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.

A Sylvester–Gallai theorem for cubic curves

We prove a variant of the Sylvester–Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in R2 is not contained in a cubic, then there is a cubic that contains exactly nine of the points. This resolves the first unknown case of a conjecture of Wiseman and Wilson from 1988, who proved a variant of Sylvester–Gallai for conics and conjectured that similar statements hold for curves of any degree.

A Rarity in Geometry

We study the locus C of all points in the plane whose pedal points on the six sides of a complete quadrangle lie on a conic. In the Euclidean plane, it turns out that C is an algebraic curve of degree 7 and genus 5 and not of degree 12 as it could be expected. Septic curves occur rather seldom in geometry which motivates a detailed study of this particular curve. We look at its singularities, focal points, and those points on C whose pedal conics degenerate. Then, we show that the septic curve occurs as the locus curve for a more general question. Further, we describe those cases where C degenerates or is of degree less than 7 depending on the shape of the initial quadrilateral

Piecewise linear differential systems with an algebraic line of separation

We study the number of limit cycles of planar piecewise linear differential systems separated by a branch of an algebraic curve. We show that for each \(n\in\mathbb{N}\) there exist piecewise linear differential systems separated by an algebraic curve of degree \(n\) having [n/2] hyperbolic limit cycles. Moreover, when n=2,3, we study in more detail the problem, considering a perturbation of a center and constructing examples with 4 and 5 limit cycles, respectively. These results follow by proving that the set of functions generating the first order averaged function associated to the problem is an extended complete Chebyshev system in a suitable interval.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/19/abstr.html

Zeroes and rational points of analytic functions

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the graph $\Gamma_f$ of $f_{\vert D}$ and algebraic curves of degree $d$ is polynomially bounded in $d$. In particular, we show these assumptions are satisfied for random power series, for some explicit classes of lacunary series, and for solutions of linear differential equations with coefficients in ${\bf Q}[z]$. As a consequence, for any function $f$ in these families, $\Gamma_f$ has less than $\beta \log^\alpha T$ rational points of height at most $T$, for some $\alpha, \beta >0$.

Quadratic systems with a symmetrical solution

In this paper we study the existence and uniqueness of limit cycles for so-called quadratic systems with a symmetrical solution: dx(t)/dt = P2(x, y) ≡ a00+ a10x + a01y + a20x2+ a11xy + a02y2dy(t)/dt = Q2(x, y) ≡ b00+ b10x + b01y + b20x2+ b11xy + b02y2where (x, y) ∈ ℝ2t ∈ ℝ, aijbij∈ ℝ, i.e. a real planar system of autonomous ordinary differential equations with linear and quadratic terms in the two independent variables. We prove that a quadratic system with a solution symmetrical with respect to a line can be of two types only. Either the solution is an algebraic curve of degree at most 3 or all solutions of the quadratic system are symmetrical with respect to this line. For completeness we give a new proof of the uniqueness of limit cycles for quadratic systems with a cubic algebraic invariant, a result previously only available in Chinese literature. Together with known results about quadratic systems with algebraic invariants of degree 2 and lower, this implies the main result of this paper, i.e. that quadratic systems with a symmetrical solution have at most one limit cycle which if it exists is hyperbolic.

Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions

Let d and k be positive integers. Let μ be a positive Borel measure on R2 possessing finite moments up to degree 2d−1. If the support of μ is contained in an algebraic curve of degree k, then we show that there exists a quadrature rule for μ with at most dk many nodes all placed on the curve (and positive weights) that is exact on all polynomials of degree at most 2d−1. This generalizes both Gauss and (the odd degree case of) Szegő quadrature where the curve is a line and a circle, respectively, to arbitrary plane algebraic curves. We use this result to show that, without any hypothesis on the support of μ, there is always a cubature rule for μ with at most 32d(d−1)+1 many nodes. In both results, we show that the quadrature or cubature rule can be chosen such that its value on a certain positive definite form of degree 2d is minimized. We characterize the unique Gaussian quadrature rule on the line as the one that minimizes this value or several other values as for example the sum of the nodes’ distances to the origin. The tools we develop should prove useful for obtaining similar results inhigher-dimensional cases although at the present stage we can present only partial results in that direction.

On Sets Defining Few Ordinary Circles

An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least n^2/4 - O(n) ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most n^3/24 - O(n^2) circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most Kn^2 ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

A bound for the Milnor number of plane curve singularities

Abstract Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

An Explanation to “Rolling Simplexes and Their Commensurability” (Field Equations in Accordance with Tycho Brahe)

Various Cartesian models of central power fields with quadratic dynamics are studied. These examples lead the reader to comprehension of basic aspects of the differential algebraic-geometrical Brahe–Descartes–Wotton theory, which embraces central power fields whose dynamics is composed of flat affine algebraic curves of degree at most N (N = 1, 2, 3, . . .). When N = 2, a quadratic rolling simplex law is proved by purely algebraic means.

Laws of rolling simplexes (feild equations according to Tycho Brahe)

Models of central power fields with quadratic dynamics suggested by Rene Descartes are studied. These examples gradually lead the reader to the understanding of base notions of differen-tial algebraic-geometric theory of Brahe-Descartes-Wotton including central fields whose dynamics is formed by planar affine algebraic curves of degree not exceeding N (N = 1, 2, 3,...).

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.

Path splines with conic envelopes

We look at path splines as constructed by joining tangent continuously segments of envelopes of 1-parameter families of conics. Each envelope is an algebraic curve of degree four or less. A path spline interpolates a sequence of point/tangent pairs and for each segment there is an additional shape handle which allows for the dilation or contraction of that segment of the path spline without altering the rest of it. The mathematical construction of the spline reduces to defining a Bézier curve in higher dimensional space: the space of conics. The cubic hypersurface corresponding to the degenerate conics plays a role in the construction.

Kempe’s linkages and the Universality Theorem

Inspired by James Watt’s approximate straight line generator, kinematicians of the 19th century challenged themselves to design a mechanical device that could convert rotary motion into a perfect straight line and vice versa. Few inventions emerged in 1864 due to Peaucellier and Lipkin and in 1875 due to Hart. Just a year later, in 1876, Alfred B Kempe presented a generalized method for linkages that could exactly trace any algebraic curve of degree n and not just a straight line. This work of Kempe is of classical importance. Yet, many are not aware of it perhaps because the resulting linkages are quite complex. This article discusses Kempe’s method that highlights the way he treated the rotations analytically using only parallelograms and contra-parallelograms to get the final rigidbody linkage tracing a given algebraic curve. An elaborate example with geometric construction using only a ruler and compass is presented to help the readers understand the assembly of Kempe’s linkages.

On the graph of a function over a prime field whose small powers have bounded degree

Let f be a function from a finite field F p with a prime number p of elements, to F p . In this article we consider those functions f ( X ) for which there is a positive integer n > 2 p − 1 − 11 4 with the property that f ( X ) i , when considered as an element of F p [ X ] / ( X p − X ) , has degree at most p − 2 − n + i , for all i = 1 , … , n . We prove that every line is incident with at most t − 1 points of the graph of f , or at least n + 4 − t points, where t is a positive integer satisfying n > ( p − 1 ) / t + t − 3 if n is even and n > ( p − 3 ) / t + t − 2 if n is odd. With the additional hypothesis that there are t − 1 lines that are incident with at least t points of the graph of f , we prove that the graph of f is contained in these t − 1 lines. We conjecture that the graph of f is contained in an algebraic curve of degree t − 1 and prove the conjecture for t = 2 and t = 3 . These results apply to functions that determine less than p − 2 p − 1 + 11 4 directions. In particular, the proof of the conjecture for t = 2 and t = 3 gives new proofs of the result of Lovász and Schrijver [L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Sci. Math. Hungar. 16 (1981) 449–454] and the result in [A. Gács, On a generalization of Rédei’s theorem, Combinatorica 23 (2003) 585–598] respectively, which classify all functions which determine at most 2 ( p − 1 ) / 3 directions.

Least Squares Fitting of Piecewise Algebraic Curves

A piecewise algebraic curve is defined as the zero contour of a bivariate spline. In this paper, we present a new method for fittingC1piecewise algebraic curves of degree 2 over type-2 triangulation to the given scattered data. By simultaneously approximating points, associated normals and tangents, and points constraints, the energy term is also considered in the method. Moreover, some examples are presented.

On the q-numerical range of matrices and matrix polynomials

The q-numerical range (0≤ q ≤ 1) of an n × n matrix polynomial is defined by In this article, we investigate the boundary and the shape of Wq (P), using the notion of local dimension. We also obtain that the q-numerical range of first-order matrix polynomials is always simply connected. Moreover, the special cases of 2×2 matrices and matrix polynomials are considered. In particular, the boundary of the q-numerical range of a 2×2 matrix polynomial of degree m lies on an algebraic curve of degree at most 8m.

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R [ h ] module by several basic integrals which satisfy the Picard–Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k + n , under polynomial perturbation of arbitrary degree.

Robust plotting of generalized lemniscates

A root neighborhood (or pseudozero set) of a degree- n polynomial p( z) is the set of roots of all polynomials p ̃ (z) whose coefficients differ from those of p( z), under a specified norm in C n+1 , by no more than a given amount ε. In the case of a weighted infinity norm, root neighborhoods are bounded by generalized lemniscates (algebraic curves of degree 4 n). A simple description of generalized lemniscates may be cast in terms of multipolar coordinates: namely, the product of the distances r 1,…, r n of a variable point z from the n roots z 1,…, z n of p( z) is equal to a degree- n polynomial in the distance r of z from the origin (in the case of “ordinary” lemniscates, this polynomial specializes to a constant). The ability to efficiently and faithfully graph lemniscates is important in numerical analysis, control theory, and other scientific/engineering applications. By constructing the tensor-product Bernstein form over a bounding rectangle, we develop a robust adaptive algorithm to plot lemniscates, employing quadtree subdivision to efficiently achieve any desired resolution. Moreover, singular points of the lemniscate (if any) may be explicitly identified and incorporated into the subdivision.