Binary Forms Of Degree Research Articles

Extreme Points of Gram Spectrahedra of Binary Forms

The Gram spectrahedron mathrm {Gram}(f) of a form f with real coefficients is a compact affine-linear section of the cone of psd symmetric matrices. It parametrizes the sum of squares decompositions of f, modulo orthogonal equivalence. For f a sufficiently general positive binary form of arbitrary degree, we show that mathrm {Gram}(f) has extreme points of all ranks in the Pataki range. We also calculate the dimension of the set of rank r extreme points, for any r. Moreover, we determine the pairs of rank two extreme points for which the connecting line segment is an edge of mathrm {Gram}(f). The proof of the main result relies on a purely algebraic fact of independent interest: Whenever d,rge 1 are integers with left( {begin{array}{c}r+1 2end{array}}right) le 2d+1, there exists a length r sequence f_1,dots ,f_r of binary forms of degree d for which the left( {begin{array}{c}r+1 2end{array}}right) pairwise products f_if_j, ile j, are linearly independent.

On the automorphism group of a binary form associated with algebraic trigonometric quantities

TextLet F(x,y) be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group Aut F for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of 2cos⁡(2πn) and 2sin⁡(2πn), which we denote by Ψn(x,y) and Πn(x,y), respectively. The remaining two forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted Tn(x,y) and Un(x,y), respectively. VideoFor a video summary of this paper, please visit https://youtu.be/1Ge-cm4ilek.

An Algebraic Approach to Electron Interactions in Quantum Hall Systems

Let m denote the number of quasielectrons (QEs) in a quantum Hall system containing N particles altogether. We show in several general cases that for systems containing m QEs in a single angular momentum shell above N − m Fermions in an incompressible quantum liquid (IQL) state having filling factor ν =1/3 that there always exists a configuration whose symmetric correlation function G is nonzero. This extends recent comparable results concerning the IQL state. As a consequence, one can obtain (explicitly) a configuration having a nonzero G for all N ≤ 8 particle systems containing any number of QEs. To establish our result, we construct a family of multi-graphs on N vertices satisfying certain restraints on the degrees of the vertices and possessing the property that whenever one computes the linear symmetrization of the graph monomial of any member of the family, the result is always nonzero. The nonzero linear symmetrization that is obtained in each case is in fact an example of what is called a relative semi-invariant of a (generic) binary form of degree N. Thus, in addition to providing new correlation functions for systems of interacting Fermions containing QEs, our construction is of potential interest from both the invariant and graph theoretic standpoints.

ON THE AREA OF THE FUNDAMENTAL REGION OF A BINARY FORM ASSOCIATED WITH ALGEBRAIC TRIGONOMETRIC QUANTITIES

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. We estimate the area $A_F$ bounded by the curve $|F(x, y)| = 1$ for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of $2\cos\left(\frac{2\pi}{n}\right)$ and $2\sin\left(\frac{2\pi}{n}\right)$, which we denote by $\Psi_n(x, y)$ and $\Pi_n(x, y)$, respectively. The remaining two families of binary forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted $T_n(x, y)$ and $U_n(x, y)$, respectively.

The monic rank

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.

On The betti numbers of oriented Grassmannians and independent semi-invariants of binary forms

AbstractWe present a complete functional formula expressing theith ℤ2-Betti number of the oriented Grassmann manifold of oriented 3-dimensional vector subspaces in Euclideann-space forifrom the range determined by the characteristic rank of the canonical oriented 3-dimensional vector bundle over this manifold. The same formula explicitly exhibits the number of linearly independent semi-invariants of degree 3 of a binary form of degreen− 3. Using the approach and data presented in this note, analogous results can be obtained for the oriented Grassmann manifold of oriented 4-dimensional vector subspaces in Euclideann-space and semi-invariants of degree 4 of a binary form of degreen− 4.

Binary forms with three different relative ranks

Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \subseteq \mathbb C$. The $K$-rank of $f$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fields. The $K$-rank of a form $f$ (such as $x^3y^2$) may depend on whether -1 is a sum of two squares in $K$.

Diophantine inequality involving binary forms

Let d ≥ 3 be an integer, and set r = 2d-1 + 1 for 3 ≤ d ≤ 4; r = 17/32∙d+1 for 5 ≤ d ≤ 6; r = d2+d+1 for 7 ≤ d ≤ 8; and r = d2+d+2 for d ≥ 9; respectively. Suppose that Фi(x, y) ∈ Z[x, y] (1 ≤ i ≤ r) are homogeneous and nondegenerate binary forms of degree d: Suppose further that λ1, λ2,..., λr are nonzero real numbers with λ1/λ2 irrational, and λ1Ф1(x1, y1) + λ2Ф2(x2, y2) +... λrФr(xr, yr) is indefinite. Then for any given real η and σ with 0 < σ < 22-d, it is proved that the inequality $$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$ has infinitely many solutions in integers x1; x2,..., xr, y1, y2,... yr: This result constitutes an improvement upon that of B. Q. Xue.

REPRESENTATION OF SMALL INTEGERS BY BINARY FORMS

We establish some upper bounds for the number of integer solutions to the Thue inequality |$|F(x , y)| \leq m$|⁠, where |$F$| is a binary form of degree |$n \geq 3$| and with non-zero discriminant |$D$|⁠, and |$m$| is an integer. Our upper bounds are independent of |$m$|⁠, when |$m$| is smaller than |$|D|^{{1}/{4(n-1)}}$|⁠. We also consider the Thue equation |$|F(x , y)| = m$| and give some upper bounds for the number of its integral solutions. In the case of equation, our upper bounds will be independent of integer |$m$|⁠, when |$m < |D|^{{1}/{2(n-1)}}$|⁠.

A CONDITION FOR CYCLIC ALGEBRAS TO BE SPLIT

A simple sufficient condition for certain cyclic algebras of odd degree d to be split is presented. It employs certain binary forms of degree d and the values they represent. A similar sufficient condition for certain Albert algebras not to be division algebras is found as well.

On totally reducible binary forms: II.

Let $f$ be a binary form of degree $l\geq3$, that is, a product of linear forms with integer coefficients. The principal result of this paper is an asymptotic formula of the shape $n^{2/l}(C(f)+O(n^{-\eta_l+\varepsilon}))$ for the number of positive integers not exceeding $n$ that are representable by $f$; here $C(f)&gt;0$ and $\eta_l&gt;0$.

A Classification of Non-Degenerated Binary Forms

There is given a representation of the equivalence classes of non-degenerated binary forms of degree d as quotient of an affine open subset of (d – 3)-dimensional affine space by the d-th symmetric group.

Minimum number of fundamental invariants for the binary form of degree 7

The minimal number of fundamental invariants for the binary form of degree 7 was a problem left open since last century. It has been solved partly by computer algebra, partly by hand computations.

Thue Equations with Few Coefficients

Let $F(x, y)$ be a binary form of degree $r \geqslant 3$ with integer coefficients, and irreducible over the rationals. Suppose that only $s + 1$ of the $r + 1$ coefficients of $F$ are nonzero. Then the Thue equations $F(x, y) = 1$ has $\ll {(rs)^{1/2}}$ solutions. More generally, the inequality $|F(x, y)| \leqslant h$ has $\ll {(rs)^{1/2}}{h^{2/r}}(1 + \log {h^{1/r}})$ solutions.

On representation of numbers by binary forms

An effective method is given for finding all rational points, the denominators of which are formed from a finite number of fixed primes, on the curvef (x, y)=A, wheref (x, y) is a binary form of degree three at least, irreducible over the field of rational numbers, and A is a rational number.