Related papers: The Newton-Puiseux algorithm and effective algebra…
We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \pi, in Logspace. Our main result is that…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
A molecule is a nonzero non-unit element of an integral domain (resp., commutative cancellative monoid) having a unique factorization into irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as well as the molecules…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
Let p be a prime number. In this paper we use an old technique of Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by a p-regular equation. To illustrate the potential applications of…
Let $S\subset R^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining $S$ and…
Solving a decision theory problem usually involves finding the actions, among a set of possible ones, which optimize the expected reward, possibly accounting for the uncertainty of the environment. In this paper, we introduce the…
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of…
We present a new algorithm for computing integral bases in algebraic function fields of one variable, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of the concepts of localization and…
We describe an algorithm, implemented in Python, which can enumerate any permutation class with polynomial enumeration from a structural description of the class. In particular, this allows us to find formulas for the number of permutations…
A relationship between Puiseux series satisfying an ordinary differential equation corresponding to a polynomial dynamical system and degrees of irreducible invariant algebraic curves is studied. A bound on the degrees of irreducible…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
We develop a search algorithm for systems of $q$-difference equations satisfied by Andrews-Gordon type double series. We then couple the search algorithm with Euler's algorithm for finding infinite products to narrow the search space. We…
We establish an effective improvement on the Liouville inequality for approximation to complex non-real algebraic numbers by quadratic complex algebraic numbers.
Let $F(t,u)\equiv F(u)$ be a formal power series in $t$ with polynomial coefficients in $u$. Let $F\_1, ..., F\_k$ be $k$ formal power series in $t$, independent of $u$. Assume all these series are characterized by a polynomial equation $$…
Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly…
We propose a heuristic algorithm for fast computation of the Poincar\'{e} series $P_n(t)$ of the invariants of binary forms of degree $n$, viewed as rational functions. The algorithm is based on certain polynomial identities which remain to…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.
In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems)…