In the paper we prove that a multivariate polynomial has algebraically dependent roots iff the coefficients are algebraic numbers up to a common proportional term. A complex analytic proof can be found in [2] with applications in the theory of linear functional equations, see also [3, an open problem, section 4.4] and [1]. Here we present an elementary proof involving cardinality properties and...

For any partition λ of an integer n , we write λ =< 11, 22, . . . , nn > where mi(λ) is the number of parts equal to i . We denote by r(λ) the number of parts of λ (i.e. r(λ) = ∑n i=1mi(λ) ). Recall that the notation λ ` n means that λ is a partition of n . For 1 ≤ k ≤ N , let ek be the k-th elementary symmetric function in the variables x1, . . . , xN , let hk be the sum of all monomials of to...

Let A = F [x1, x2, . . . , xn] be a ring of polynomials over a field F on the variables x1, x2, . . . , xn. It is well known (see, for example, [11]) that the study of automorphisms of the algebra A is closely related with the description of its subalgebras. By the theorem of P. M. Cohn [4], a subalgebra of the algebra F [x] is free if and only if it is integrally closed. The theorem of A. Zaks...

We study invariants of three-qubit states under local unitary transformations, i.e. functions on the space of entanglement types, which is known to have dimension 6. We show that there is no set of six algebraically independent polynomial invariants of degree ≤ 6, and find such a set with maximum degree 8. We describe an intrinsic definition of a canonical state on each orbit, and discuss the (...

Proof. Let A = k[s1, . . . , sn] where s1, . . . , sn ∈ A. We assume that A is not integral over k, in which case at least one of the si is not algebraic over k. If the set {s1, . . . , sn} is algebraically independent, then we are done. Otherwise assume that sn is algebraic over k[s1, . . . , sn−1] (by relabeling if necessary). Let f(x1, . . . , xn) be a nonzero polynomial with f(s1, . . . , s...

Results are proved indicating that the Veronese map vd often increases independence of both sets of points and sets of subspaces. For example, any d + 1 Veronesean points of degree d are independent. Similarly, the dth power map on the space of linear forms of a polynomial algebra also often increases independence of both sets of points and sets of subspaces. These ideas produce d+ 1-independen...

This papers investigates the manipulation of statements of strong independence in probabilistic logic. Inference methods based on polynomial programming are presented for strong independence, both for unconditional and conditional cases. We also consider graph-theoretic representations, where each node in a graph is associated with a Boolean variable and edges carry a Markov condition. The resu...

Naive Bayes is a simple Bayesian classifier with strong independence assumptions among the attributes. This classifier, despite its strong independence assumptions, often performs well in practice. It is believed that relaxing the independence assumptions of a naive Bayes classifier may improve the classification accuracy of the resulting structure. While finding an optimal unconstrained Bayesi...

In this paper, we discuss stability and linear independence of the integer translates of a scaling vector = (1 ; ; r) T , which satisses a matrix reenement equation (x) = n X k=0 P k (2x ? k); where (P k) is a nite matrix sequence. We call P (z) = 1 2 P P k z k the symbol of. Stable scaling vectors often serve as generators of multiresolution analyses (MRA) and therefore play an important role ...

iff px ∈ [0, 1) for all x. When the events are not independent, an answer is given by the Lovász local lemma (and its variations). Definition 14.1. We say that G is the dependency graph of (Ax)x∈X if for all x ∈ X, Ax is independent of the σ-algebra generated by the collection {Ay | y / ∈ Γ∗(x)} (where Γ(x) is the set of neighbours of x in G and Γ∗(x) := Γ(x) ∪ {x}). Theorem 14.2 (Lovász local ...