Related papers: A note on dimensions of polynomial size circuits

This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its…

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp(L) is the set of all effective Hausdorff dimensions of individual…

We show that there is a defining equation of degree at most $\mathsf{poly}(n)$ for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field $\mathbb{F}$, there is a non-zero…

We study problems related to indecomposability of modules over certain local finite dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial…

In this paper we generalize the Ritt-Kolchin method of characteristic sets and the classical Gr\"obner basis technique to prove the existence and obtain methods of computation of multivariate difference-differential dimension polynomials…

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations…

We have considered phi^4 theory in higher dimensions. Using functional diagrammatic approach, we computed the one-loop correction to effective potential of the scalar field in five dimensions. It is shown that phi^4 theory can be…

Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $\epsilon=\epsilon(\mathbb{P})>0$ such that, for any compact set $E \subset…

We introduce a class of polynomial maps that we call polynomial roots of powerseries, and show that automorphisms with this property generate the automorphism group in any dimension. In particular we determine generically which polynomial…

The massive field theory approach in fixed space dimensions $d=3$ is applied to investigate a dilute solution of long-flexible polymer chains in a good solvent between two parallel repulsive walls, two inert walls and for the mixed case of…

Let $(S, m)$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $I$ be an ideal of $S$. We prove that if $\text{depth} S/I \ge 3$, then the cohomological dimension $\mathrm{cd}(S, I)$ of $I$ is less…

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side…

Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…

After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the…

We develop dimension theory for a large class of structures called espaliers, consisting of a set $L$ equipped with a partial order $\leq$, an orthogonality relation $\perp$, and an equivalence relation $\sim$, subject to certain axioms.…

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

We provide a formula and a generating function for the dimension of the vector space of relative symmetric polynomials of D_n for all its irreducible one-dimensional and two-dimensional representations which were defined originally by M.…

In this paper we study (continuous) polynomials $p: J\to X$, where $J$ is an abelian topological semigroup and $X$ is a topological vector space. If $J$ is a subsemigroup with non-empty interior of a locally compact abelian group $G$ and…

We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of type $A_n$, $B_n$, and $D_n$, the coefficients of the characteristic polynomial of the reflection…

Bonamy et al \cite{BBEGLPS} showed that graphs of polynomial growth have finite asymptotic dimension. We refine their result showing that a graph of polynomial growth strictly less than $n^{k+1}$ has asymptotic dimension at most $k$. As a…