A polynomial over a ring is called decomposable if it composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number polynomials with integer coefficients fixed degree bounded height. Moreover, asymptotic formulas monic even degree. For example, sextic which are height at most H to $$(16\zeta (3)-5/4)H^3$$ as $$H \rightarrow \infty $$ .

We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers independent linear forms, or equivalently, decomposing symmetric tensors into rank-1 linearly vectors. The rely on two facets higher degree namely Harrison's algebraic theory some algebro-geometric properties. proposed are based purely solving quadratic equations. Moreover, a byproduct our one can...

The Umehara algebra is studied with motivation on the problem of non-existence common complex submanifolds. In this paper, we prove some new results in and obtain applications. particular, if a manifolds admits holomorphic polynomial isometric immersion to one indefinite space form, then it cannot another form different type. Other consequences include submanifolds for projective or hyperbolic ...

We study consistency of learning algorithms for a multi-class performance metric that is anon-decomposable function of the confusion matrix of a classifier and cannot be expressed asa sum of losses on individual data points; examples of such performance metrics include themicro and macro F-measure used widely in information retrieval and the multi-class G-meanmetric popular in c...

Améndola et al. proposed a method for solving systems of polynomial equations lying in family which exploits recursive decomposition into smaller systems. A admits such if and only the corresponding Galois group is imprimitive. When imprimitive, we consider problem computing an explicit decomposition. consequence Esterov’s classification sparse with imprimitive groups that this obtained by insp...

Since the celebrated work of Jerrum, Sinclair, and Vigoda, we have known that the permanent of a {0, 1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain to sample perfect matchings of a bipartite graph. A separate strand of the literature has pursued the possibility of an alternate, algebraic polynomial-time approximation scheme. These schemes work...

Although the equivalence of two Ordered Binary Decision Diagrams (OBDDs) can be decided in polynomial time, the equivalence of two Free Binary Decision Diagrams (FBDDs) is only known to be probabilistically decidable in polynomial time. FBDDs are a strict superset of OBDDs, and are more succinct than OBDDs, which explains the interest in testing their equivalence. We show that the probabilistic...

This paper introduces the ulmods, a generalisation of modules for the homogeneous relations. We first present some properties of the ulmod family, then show that, if the homogeneous relation fulfills some natural axioms, the ulmod family has a unique decomposition tree. Under a certain size assumption we show that this tree can be computed in polynomial time. We apply this theory to a new tourn...

This paper introduces the ulmods, a generalisation of modules for the homogeneous relations. We first present some properties of the ulmod family, then show that, if the homogeneous relation fulfills some natural axioms, the ulmod family has a unique decomposition tree. Under a certain size assumption we show that this tree can be computed in polynomial time. We apply this theory to a new tourn...

Computing a decomposition of a polynomial f(x) as a functional composition g(h(x)) of polynomials g(x) and h(x), is an important and well-studied problem, both for exact and approximate inputs. In this paper, we re-examine the original (exponential-time) algorithm of Barton and Zippel for this task, which looks for special factors of an associated separated bivariate polynomial. We demonstrate ...