Related papers: Graphical patterns in quadratic residues

We construct a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and prove a simple…

The use of quadratic residues to construct matrices with specific determinant values is a familiar problem with connections to many areas of mathematics and statistics. Our research has focused on using cubic residues to construct matrices…

Using a quadratic version of the Bott residue theorem, we give a quadratic refinement of the count of twisted cubic curves on hypersurfaces and complete intersections in a projective space.

Let $D$ be an irreducible plane curve. In this article, we first introduce a notion of a quadratic residue curve mod $D$, and study quadratic residue concis $C$ mod an irreducible quartic curve $Q$. As an application, we study a dihedral…

The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi's…

In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components.

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…

The paper presents mathematical models of quasicrystals with particular attention given to cut-and-project sets. We summarize the properties of higher-dimensional quasicrystal models and then focus on the one-dimensional ones. For the…

Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.

A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.

We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.

We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…

In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results we construct algebraic-geometric codes which…

In this note, we prove that under some conditions, certain products of integers related to Gauss factorials are always quadratic residues.

In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and…

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.

When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive $\ell$-tuples of quadratic residues, studying corresponding algebraic curves and their…

We give some results on quadratic normality of reducible curves canonically embedded and partially extend this study to their projective normality.

These are the notes from my courses on the arithmetic of quadratic forms.