The homogeneous ideals of curves in a double plane have been studied by Chiarli, Greco, Nagel. Completing this work we describe the equations of any curve that is contained in some quadric. As a consequence, we classify the Hartshorne-Rao modules of such curves.

Let $A=K[x_1,\cdots ,x_m]$ be a polynomial ring in $m$ variables and let $\mathbf{d}=(d_0\<\cdots \<d_m)$ be a strictly increasing sequence of $m+1$ integers. Boij and Söderberg conjectured the existence of graded $A$-modules $M$ of finite length having pure free resolution of type $\mathbf{d}$ in the sense that for $i=0,\cdots ,m$ the $i$-th syzygy module of $M$ has generators only in degree $d_i$.This paper provides a construction, in characteristic zero, of modules with this property that are also $GL\left(m\right)$-equivariant. Moreover, the construction works over rings of the form $A{\otimes }_{K}B$ where $A$ is a polynomial...

In this short note, we extend Faugére’s F4-algorithm for computing Gröbner bases to polynomial rings with coefficients in an Euclidean ring. Instead of successively reducing single S-polynomials as in Buchberger’s algorithm, the F4-algorithm is based on the simultaneous reduction of several polynomials.

Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial

We give a description of all polynomial constants of the five-variable Volterra derivation, hence of all polynomial first integrals of its corresponding Volterra system of differential equations. The Volterra system plays a significant role in plasma physics and population biology.

The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras $\mathrm{Lie}{G}_2\stackrel{i}{\hookrightarrow }so\left(7\right)$, and generalized conformal $so\left(7\right)$-Verma modules of scalar type. As a result, we classify the $i\left(\mathrm{Lie}{G}_2\right)\cap 𝔭$-singular vectors for this class of $so\left(7\right)$-modules.

We present some facts, observations and remarks concerning the problem of finiteness of the rings of constants for derivations of polynomial rings over a commutative ring k containing the field ℚ of rational numbers.

For a finite commutative ring $R$ and a positive integer $k⩾2$, we construct an iteration digraph $G(R,k)$ whose vertex set is $R$ and for which there is a directed edge from $a\in R$ to $b\in R$ if $b=a^k$. Let $R=R_1\oplus ...\oplus R_s$, where $s>1$ and $R_i$ is a finite commutative local ring for $i\in \{1,...,s\}$. Let $N$ be a subset of $\{R_1,\cdots ,R_s\}$ (it is possible that $N$ is the empty set $\varnothing$). We define the fundamental constituents $G_N^{*}(R,k)$ of $G(R,k)$ induced by the vertices which are of the form $\{(a_1,\cdots ,a_s)\in R:a_i\in \mathrm{D}\left(R_i\right)$ if $R_i\in N$, otherwise $a_i\in \mathrm{U}\left(R_i\right),i=1,...,s\},$ where U$\left(R\right)$ denotes the unit group of $R$ and D$\left(R\right)$ denotes the zero-divisor set of $R$. We investigate...