We show that the problem of constructing telescopers for functions of m
variables is equivalent to the problem of constructing telescopers for
algebraic functions of m -1 variables and present a new algorithm to construct
telescopers for algebraic functions of two variables. These considerations are
based on analyzing the residues of the input. According to experiments, the
resulting algorithm for rational functions of three variables is faster than
known algorithms, at least in some examples of combinatorial interest.

Loop invariants play a very important role in proving correctness of
programs. In this paper, we address the problem of generating invariants of
polynomial loop programs. We present a new approach, for generating polynomial
equation invariants of polynomial loop programs through computing vanishing
ideals of sample points. We apply rational function interpolation, based on
early termination technique, to generate invariants of loop programs with
symbolic initial values. Our approach avoids first-order quantifier elimination
and cylindrical algebraic decomposition(CAD).

The purpose of this paper is twofold. An immediate practical use of the
presented algorithm is its applicability to the parametric solution of
underdetermined linear ordinary differential equations (ODEs) with coefficients
that are arbitrary analytic functions in the independent variable. A second
conceptual aim is to present an algorithm that is in some sense dual to the
fundamental Euclids algorithm, and thus an alternative to the special case of a
Groebner basis algorithm as it is used for solving linear ODE-systems.

Lazard and Rouillier in [9], by introducing the concept of discriminant
variety, have described a new and efficient algorithm for solving parametric
polynomial systems. In this paper we modify this algorithm, and we show that
with our improvements the output of our algorithm is always minimal and it does
not need to compute the radical of ideals.

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and
the Moore-Penrose inverse of a given rational matrix A is established. Classes
A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R*
and T*(AT*)+, where R and T are rational matrices with appropriate dimensions
and corresponding rank. The proposed algorithm is based on these general
representations and the Cholesky factorization of symmetric positive matrices.
The algorithm is implemented in programming languages MATHEMATICA and DELPHI,
and illustrated via examples.

We introduce a method and an algorithm for computing the weighted
Moore-Penrose inverse of multiple-variable polynomial matrix and the related
algorithm which is appropriated for sparse polynomial matrices. These methods
and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S.
Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose
inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to
multiple-variable rational and polynomial matrices and improvements of these
algorithms on sparse matrices.

In this paper, a multiplicity preserving triangular set decomposition
algorithm is proposed for a system of two polynomials. The algorithm decomposes
the variety defined by the polynomial system into unmixed components
represented by triangular sets, which may have negative multiplicities. In the
bivariate case, we give a complete algorithm to decompose the system into
multiplicity preserving triangular sets with positive multiplicities. We also
analyze the complexity of the algorithm in the bivariate case.

This paper presents a conception for computing gr\"{o}bner basis. We convert
some of gr\"{o}bner-computing algorithms, e.g., F5, extended F5 and GWV
algorithms into a special type of algorithm. The new algorithm's finite
termination problem can be described by equivalent conditions, so all the above
algorithms can be determined when they terminate finitely. At last, a new
criterion is presented. It is an improvement for the Rewritten and Signature
Criterion.

Efficient characteristic set methods for computing solutions of polynomial
equation systems in a finite field are proposed. The concept of proper
triangular sets is introduced and an explicit formula for the number of
solutions of a proper and monic (or regular) triangular set is given. An
improved zero decomposition algorithm which can be used to reduce the zero set
of an equation system in general form to the union of zero sets of monic proper
triangular sets is proposed. As a consequence, we can give an explicit formula
for the number of solutions of an equation system.

A new efficient algorithm is proposed for factoring polynomials over an
algebraic extension field. The extension field is defined by a polynomial ring
modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no
extra Groebner basis computation is needed for factoring a polynomial over this
extension field. Nothing more than linear algebraic technique is used to get a
polynomial over the ground field by a generic linear map. Then this polynomial
is factorized over the ground field.

We give a new algorithm to find local maximum and minimum of a holonomic
function and apply it for the Fisher-Bingham integral on the sphere $S^n$,
which is used in the directional statistics. The method utilizes the theory and
algorithms of holonomic systems.

We provide a "shared axiomatization" of natural numbers and hereditarily
finite sets built around a polymorphic abstraction of bijective base-2
arithmetics.

The "axiomatization" is described as a progressive refinement of Haskell type
classes with examples of instances converging to an efficient implementation in
terms of arbitrary length integers and bit operations. As an instance, we
derive algorithms to perform arithmetic operations efficiently directly with
hereditarily finite sets.

In this paper we derive aggregate separation bounds, named after
Davenport-Mahler-Mignotte (\dmm), on the isolated roots of polynomial systems,
specifically on the minimum distance between any two such roots. The bounds
exploit the structure of the system and the height of the sparse (or toric)
resultant by means of mixed volume, as well as recent advances on aggregate
root bounds for univariate polynomials, and are applicable to arbitrary
positive dimensional systems.

We present an efficient solution to the following problem, of relevance in a
numerical optimization scheme: calculation of integrals of the type \[\iint_{T
\cap \{f\ge0\}} \phi_1\phi_2 \, dx\,dy\] for quadratic polynomials
$f,\phi_1,\phi_2$ on a plane triangle $T$. The naive approach would involve
consideration of the many possible shapes of $T\cap\{f\geq0\}$ (possibly after
a convenient transformation) and parameterizing its border, in order to
integrate the variables separately.

In this paper we present two algorithms for the multiplication of sparse
Laurent polynomials and Poisson series (the latter being algebraic structures
commonly arising in Celestial Mechanics from the application of perturbation
theories). Both algorithms first employ the Kronecker substitution technique to
reduce multivariate multiplication to univariate multiplication, and then use
the schoolbook method to perform the univariate multiplication.

Insa and Pauer presented a basic theory of Groebner basis for differential
operators with coefficients in a commutative ring in 1998, and a criterion was
proposed to determine if a set of differential operators is a Groebner basis.
In this paper, we will give a new criterion such that Insa and Pauer's
criterion could be concluded as a special case and one could compute the
Groebner basis more efficiently by this new criterion.

In this paper, we mainly study the robust stability of linear continuous
systems with parameter uncertainties, a more general kind of uncertainties for
system matrices is considered, i.e., entries of system matrices are rational
functions of uncertain parameters which are varying in intervals. we present a
method which can check the robust Hurwitz stability of such uncertain systems
in finite steps. Examples show the efficiency of our approach.

Regular chains and triangular decompositions are fundamental and
well-developed tools for describing the complex solutions of polynomial
systems. This paper proposes adaptations of these tools focusing on solutions
of the real analogue: systems of equations, inequations and inequalities given
by polynomials with rational number coefficients.

We present a lattice algorithm specifically designed for some classical
applications of lattice reduction. The applications are for lattice bases with
a generalized knapsack-type structure, where the target vectors are boundably
short. For such applications, the complexity of the algorithm improves
traditional lattice reduction by replacing some dependence on the bit-length of
the input vectors by some dependence on the bound for the output vectors.

Solving multi-homogeneous systems, as a wide range of structured algebraic
systems occurring frequently in practical problems, is of first importance.
Experimentally, solving these systems with Gr\"obner bases algorithms seems
easier than solving homogeneous systems of the same degree, but the reasons of
this behaviour are not clear. In this paper, we focus on bilinear systems (i.e.
bihomogeneous systems where all equations have bidegree (1,1)).

We present a complete algorithm for finding an exact minimal polynomial from
its approximate value by using an improved parameterized integer relation
construction method. Our result is superior to the existence of error
controlling on obtaining an exact rational number from its approximation. The
algorithm is applicable for finding exact minimal polynomial of an algebraic
number by its approximate root.

Finite-state tree automata are a well studied formalism for representing term
languages. This paper studies the problem of determining the regularity of the
set of instances of a finite set of terms with variables, where each variable
is restricted to instantiations of a regular set given by a tree automaton. The
problem was recently proved decidable, but with an unknown complexity. Here,
the exact complexity of the problem is determined by proving
EXPTIME-completeness.

We describe a new algorithm for computing exp(f) where f is a power series in
C[[x]]. If M(n) denotes the cost of multiplying polynomials of degree n, the
new algorithm costs (2.1666... + o(1)) M(n) to compute exp(f) to order n. This
improves on the previous best result, namely (2.333... + o(1)) M(n).

We prove explicit bounds on the radius of a ball centered at the origin which
is guaranteed to contain all bounded connected components of a semi-algebraic
set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving
$s$ polynomials in $\mathbbm{Z}[X_1, ..., X_k]$ having degrees at most $d$, and
whose coefficients have bitsizes at most $\tau$. Our bound is an explicit
function of $s, d, k$ and $\tau$, and does not contain any undetermined
constants.

In this paper we study the local behavior of an algebraic curve under a
geometric construction which is a variation of the usual offsetting
construction, namely the {\it generalized} offsetting process (\cite {SS99}).
More precisely, here we discuss when and how this geometric construction may
cause local changes in the shape of an algebraic curve, and we compare our
results with those obtained for the case of classical offsets (\cite{JGS07}).
For these purposes, we use well-known notions of Differential Geometry, and
also the notion of {\it local shape} introduced in \cite{JGS07}.