Related papers: Differential algebraic dependence and Novikov depe…
We prove the Freiheitssatz for Novikov algebras in characteristic zero. It is also proved that the variety of Novikov algebras is generated by a Novikov algebra on the space of polynomials $k[x]$ in a single variable $x$ over a field $k$…
We generalize and unify the proofs of several results on algebraic in- dependence of arithmetic functions and Dirichlet series by a theorem of Ax on differential Schanuel conjecture. Along the way, we find counter-examples to some results…
As it is known, the defining identities of a free Novikov algebra can be obtained from a commutative algebra with a derivation. In this paper, we consider a class of algebras obtained from the class of associative algebras with a derivation…
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…
In this paper, we first present a classification theorem of infinite-dimensional simple Novikov algebras over an algebraically closed field with characteristic 0. Then we classify all the irreducible modules of a certain…
We extend Kolchin's results on linear dependence over projective varieties in the constants, to linear dependence over arbitrary complete differential varieties. We show that in this more general setting, the notion of linear dependence…
We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of…
We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…
We answer the following long-standing question of Kolchin: given a system of algebraic-differential equations $\Sigma(x_1,\dots,x_n)=0$ in $m$ derivatives over a differential field of characteristic zero, is there a computable bound, that…
Let $K$ be a finite extension of $\mathbb{Q}_p$ that is totally ramified over $\mathbb{Q}_p$. The set $\mathcal{M}\mathcal{F}(K)$ consists of power series in $1+zK[[z]]$ that are solutions of differential operators in $K(z)[d/dz]$ equipped…
We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are…
We give an algorithm for constructing the algebraic hull of a given matrix Lie algebra in characteristic zero. It is based on an algorithm for finding integral linear dependencies of the roots of a polynomial, that is probably of…
For a finite set of homogeneous locally nilpotent derivations of the algebra of polynomials in several variables, a finite dimensionality criterion for the Lie algebra generated by these derivations is known. Also the structure of the…
We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field…
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a…
We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and $mathbb{Q}$-linear…
We define a class of quadratic differential algebras which are generated as differential graded algebras by the elements of an Euclidean space. Such a differential algebra is a differential calculus over the quadratic algebra of its…
Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials {f_1, ..., f_m} \subset \F[x_1, ..., x_n] are called algebraically independent if there is…
We consider a space of complex polynomials of degree $n\ge 3$ with $n-1$ distinguished periodic orbits. We prove that the multipliers of these periodic orbits considered as algebraic functions on that space, are algebraically independent…
We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…