Third Root Of Unity Research Articles

Inverse of \(\alpha\)-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Let \(X\) be bipartite mixed graph and for a unit complex number \(\alpha\), \(H_\alpha\) be its \(\alpha\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_\alpha\) has a hermitian inverse \(H_\alpha^{-1}\). In this paper we give a full description of the entries of \(H_\alpha^{-1}\) in terms of the paths between the vertices. Furthermore, for \(\alpha\) equals the primitive third root of unity \(\gamma\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_\gamma^{-1}\) is \(\pm 1\) diagonally similar to \(\gamma\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(\pm 1\) diagonal matrix.

The finitude of tamely ramified pro-p extensions of number fields with cyclic p-class groups

Let p be an odd prime and F be a number field whose p-class group is cyclic. Let F{q} be the maximal pro-p extension of F which is unramified outside a single non-p-adic prime ideal q of F. In this work, we study the finitude of the Galois group G{q}(F) of F{q} over F. We prove that G{q}(F) is finite for the majority of q's such that the generator rank of G{q}(F) is two, provided that for p=3, F is not a complex quartic field containing the primitive third roots of unity.

Identifiability for mixtures of centered Gaussians and sums of powers of quadratics

AbstractWe consider the inverse problem for the polynomial map that sends an ‐tuple of quadratic forms in variables to the sum of their th powers. This map captures the moment problem for mixtures of centered ‐variate Gaussians. In the first nontrivial case , we show that for any , this map is generically one‐to‐one (up to permutations of and third roots of unity) in two ranges: for and for , thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most . The first result is obtained by the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms, as described by Chiantini and Ottaviani [SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018–1037], while the second result is accomplished using the link between secant nondefectivity with identifiability, proved by Casarotti and Mella [J. Eur. Math. Soc. (JEMS) (2022)]. The latter approach also generalizes to sums of th powers of ‐forms for and .

Cubic Equations Through the Looking Glass of Sylvester

Summary One can hardly believe that there is still something to be said about cubic equations. To dodge this doubt, we will instead try and say something about Sylvester. He doubtless found a way of solving cubic equations. As mentioned by Rota, it was the only method in this vein that he could remember. We realize that in the generic case Sylvester’s magnificent approach aimed at reduced cubic equations boils down to an easy identity expressing a cubic polynomial as a sum of two third powers of linear forms. This leads to Cardano’s formula for cubic equations involving the third roots of unity.

A new determinant for the Q-enumeration of alternating sign matrices

Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate q, actually gives the (q−1+2+q)-enumeration of alternating sign matrices. By evaluating a generalisation of this determinant we are able to reprove a conjecture of Mills, Robbins and Rumsey stating that the Q-enumeration is a product of two polynomials in Q. Further we provide a closed product formula for the generalised determinant in the 0-, 1-, 2- and 3-enumeration case, leading to new proofs of the 1-, 2- and 3-enumeration of alternating sign matrices, and a factorisation in the 4-enumeration case. Finally we relate the 1-enumeration case of our generalised determinant to the determinant evaluations of Ciucu, Eisenkölbl, Krattenthaler and Zare, which count weighted cyclically symmetric lozenge tilings of a hexagon with a triangular hole and are a generalisation of a famous result by Andrews. As a result, we obtain alternative proofs of their determinantal evaluations using the Desnanot-Jacobi identity (Dodgson condensation).

On a conjecture of Lemmermeyer

Let [Formula: see text] be a prime and denote by [Formula: see text] a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about [Formula: see text]-class groups of pure cubic fields [Formula: see text] and of their normal closures [Formula: see text]. The main goal of this paper is to reduce Lemmermeyer’s conjecture to a problem of unit theory by showing that the conjecture of Lemmermeyer follows from Conjecture 2.9.

3-rank of ambiguous class groups of cubic Kummer extensions

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(\zeta_3)$ with $d>1$ a cube-free integer and $\zeta_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma)}$ the $3$-group of ambiguous classes of the extension $k/k_0$ with relative group $G=\operatorname{Gal}(k/k_0)=\langle\sigma\rangle$. The aims of this paper are to characterize all extensions $k/k_0$ with cyclic $3$-group of ambiguous classes $C_{k,3}^{(\sigma)}$ of order $3$, to investigate the multiplicity $m(f)$ of the conductors $f$ of these abelian extensions $k/k_0$, and to classify the fields $k$ according to the cohomology of their unit groups $E_{k}$ as Galois modules over $G$. The techniques employed for reaching these goals are relative $3$-genus fields, Hilbert norm residue symbols, quadratic $3$-ring class groups modulo $f$, the Herbrand quotient of $E_{k}$, and central orthogonal idempotents. All theoretical achievements are underpinned by extensive computational results.

Towers of Solutions of qKZ Equations and Their Applications to Loop Models

Cherednik’s type A quantum affine Knizhnik–Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for V_n-valued meromorphic functions on a complex n-torus, with V_n a module over the mathrm{GL}_n-type extended affine Hecke algebra {mathcal {H}}_n. The family ({mathcal {H}}_n)_{nge 0} of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms {mathcal {H}}_nrightarrow {mathcal {H}}_{n+1}, in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper, we consider qKZ towers (f^{(n)})_{nge 0} of solutions, which consist of twisted-symmetric polynomial solutions f^{(n)} (nge 0) of the qKZ equations that are compatible with the tower structure on ({mathcal {H}}_n)_{nge 0}. The compatibility is encoded by the so-called braid recursion relations: f^{(n+1)}(z_1,ldots ,z_{n},0) is required to coincide up to a quasi-constant factor with the push-forward of f^{(n)}(z_1,ldots ,z_{n}) by an intertwiner mu _{n}{:},V_{n}rightarrow V_{n+1} of {mathcal {H}}_{n}-modules, where V_{n+1} is considered as an {mathcal {H}}_{n}-module through the tower structure on ({mathcal {H}}_n)_{nge 0}. We associate with the dense loop model on the half-infinite cylinder with nonzero loop weights, a qKZ tower (f^{(n)})_{nge 0} of solutions. The solutions f^{(n)} are constructed from specialized dual non-symmetric Macdonald polynomials with specialized parameters using the Cherednik–Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, f^{(n)} coincides with the (suitably normalized) ground state of the inhomogeneous dense O(1) loop model on the half-infinite cylinder with circumference n.

The [formula omitted] orbifold, mirror symmetry, and Hodge structures of Calabi–Yau type

Starting from the Kähler moduli space of the rigid orbifold Z=E3∕Z3 one would expect for the cohomology of the generalized mirror to be a Hodge structure of Calabi–Yau type (1,9,9,1). We show that such a structure arises in a natural way from rational Hodge structures on Λ3Q[ω]6, where ω is a primitive third root of unity. We do not try to identify an underlying mirror geometry, but we show how special geometry arises in our abstract construction. We also show how such Hodge structure can be recovered as a polarized substructure of a bigger Hodge structure given by the third cohomology group of a six-dimensional abelian variety of Weil-type. Moreover, we recover a result of Zheng Zhang on the associates variation of Hodge structure.

Construction of the Outer Automorphism of {mathcal {S}}_{6} via a Complex Hadamard Matrix

We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers.

A Cubic Hamiltonian System with Meromorphic Solutions

A Hamiltonian system in two dependent variables is presented with properties analogous to the Hamiltonian systems associated with the six Painleve equations. Its solutions are meromorphic functions in the complex plane having only simple poles with three possible residues given by the third roots of unity. Like the Painleve equations $$P_\mathrm{II}$$ – $$P_\mathrm{VI}$$ the system has families of rational solutions that can be obtained by applying Backlund transformations to certain seed solutions.

The Diophantine Equation ax6+ by3+ cz2= 0 in Gaussian Integers

In this article, we will examine the Diophantine equation , for arbitrary rational integers a, b, and c in Gaussian integers and find all the solutions of this equation for many different values of a, b, and c. Moreover, two equations of the type , and are also discussed, where i is the imaginary unit and w is a third root of unity.

Maps preserving the local spectrum of triple product of operators

Let denote the algebra of all bounded linear operators on a Banach space . Let and be infinite-dimensional complex Banach spaces, and fix two nonzero vectors and . Then we show that a surjective map satisfiesif and only if there exists a bijective mapping such that andwhere is a third root of unity.

Rational solutions of certain Diophantine equations involving norms

In this note we present some results concerning the unirationality of the algebraic variety $\cal{S}_{f}$ given by the equation \begin{equation*} N_{K/k}(X_{1}+\alpha X_{2}+\alpha^2 X_{3})=f(t), \end{equation*} where $k$ is a number field, $K=k(\alpha)$, $\alpha$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\in k[x]$ and $f\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\in k\setminus k^{3}$. We prove that if $\op{deg}f=4$ and the variety $\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\neq 0$, then $\cal{S}_{f}$ is $k$-unirational. A similar result is proved for a broad family of quintic polynomials $f$ satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $\cal{S}_{f}$ (with non-trivial $k$-rational point) is proved for any polynomial $f$ of degree 6 with $f$ not equivalent to the polynomial $h$ satisfying the condition $h(t)\neq h(\zeta_{3}t)$, where $\zeta_{3}$ is the primitive third root of unity. We are able to prove the same result for an extension of degree 3 generated by the root of polynomial $h(x)=x^3+ax+b\in k[x]$, provided that $f(t)=t^6+a_{4}t^4+a_{1}t+a_{0}\in k[t]$ with $a_{1}a_{4}\neq 0$.

Three-Phase Barker Arrays

A 3-phase Barker array is a matrix of third roots of unity for which all out-of-phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two-dimensional 3-phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3-phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double-exponentially growing arithmetic function T such that no 3-phase Barker array of size with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3-phase Barker array of size exists, then .

Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3

The maximal isotropic subspaces in split Cayley algebras were classified by van der Blij and Springer in Nieuw Archief voor Wiskunde VIII(3):158–169, 1960. Here we translate this classification to arbitrary composition algebras. We study intersection properties of such spaces in a symmetric composition algebra, and prove two triality results: one for two-D isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. We bound the distance between isotropic spaces of various dimensions, and study the strong orthogonality relation on isotropic vectors, with its own bound on the distance. The results are used to classify maximal p-central subspaces in central simple algebras of degree p = 3. We prove various linkage properties of maximal p-central spaces and p-central elements. Analogous results are obtained for symmetric p-central elements with respect to an involution of the second kind inverting a third root of unity.

A CONDITION FOR CYCLIC ALGEBRAS TO BE SPLIT

(cf. for instance [1, 30.6, p. 415]). This criterion translates the question of whether A is split into the problem whether or not a certain element is represented by a homogeneous form (the reduced norm of l/k) of degree d in d indeterminates. In order to give a sufficient criterion for A to be split it is sometimes enough, however, to check a much simpler?binary?form of degree d that arises by restricting the reduced norm of the algebra A to a suitable subspace of A: let & be a field of characteristic not 3 that contains a primitive third root of unity, and let A = (I, a) be a cyclic algebra over k with / = k[x]/(x3 ? b) cubic etale. If the binary cubic form (a, b) represents 1 or 62, or if the binary cubic form (a, b2) represents 1 or 6, then A splits, i.e. A = Mat3(A:). More generally, take any odd integer d and let k be a field of characteristic not dividing d containing a primitive dth root of unity. Suppose that A = (/, a) is a cyclic central simple algebra over k of degree d with / = k[x]/(xd ? b), a field extension of k. Then A splits if there is an integer r, 1 < r < d ? 1, such that the form (a, br) of degree d represents bs for some integer s with r^s, 0<s<d ? 1. Also for certain 27-dimensional exceptional simple cubic Jordan algebras it suffices to check a much simpler 6-dimensional cubic form to see if the algebra

Linear Independence of q‐Logarithms over the Eisenstein Integers

For fixed complex q with |q | &gt; 1, the q‐logarithm Lq is the meromorphic continuation of the series ∑n&gt;0zn/(qn − 1), |z | &lt;|q|, into the whole complex plane. If K is an algebraic number field, one may ask if 1, Lq(1), Lq(c) are linearly independent over K for q, c ∈ K× satisfying |q | &gt; 1, c ≠ q, q2, q3, …. In 2004, Tachiya showed that this is true in the Subcase K = ℚ, q ∈ ℤ, c = −1, and the present authors extended this result to arbitrary integer q from an imaginary quadratic number field K, and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if K is the Eisenstein number field , q an integer from K, and c a primitive third root of unity. Under these conditions, the linear independence holds also for 1, Lq(c), Lq(c−1), and both results are quantitative.

Quantum Incompressibility and Razumov Stroganov Type Conjectures

We establish a correspondence between polynomial representations of the Temperley and Lieb algebra and certain deformations of the Quantum Hall Effect wave functions. When the deformation parameter is a third root of unity, the representation degenerates and the wave functions coincide with the domain wall boundary condition partition function appearing in the conjecture of A.V. Razumov and Y.G. Stroganov. In particular, this gives a proof of the identification of the sum of the entries of a O(n) transfer matrix eigenvector and a six vertex-model partition function, alternative to that of P. Di Francesco and P. Zinn-Justin.

Efficient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers

We present simple and efficient algorithms for computing the gcd and cubic residuosity in the ring of Eisenstein integers, Z [ ζ ] , i.e. the integers extended with ζ , a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O ( n 2 ) for n -bit input. For the cubic residuosity problem this is an improvement from the known results based on the Euclidean algorithm, and taking time O ( n ⋅ M ( n ) ) , where M ( n ) denotes the complexity of multiplying n -bit integers. For the gcd problem our algorithm is simpler and faster than an earlier algorithm of complexity O ( n 2 ) . The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols.