Evaluating geometric queries using few arithmetic operations

Let \cp:=(P_1,...,P_s) be a given family of $n$-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by $d$ and $h$, respectively. Suppose furthermore that for each $1\leq i\leq s$ the polynomial $P_i$ can be evaluated using $L$ arithmetic operations (additions, subtractions, multiplications and the constants 0 and 1). Assume that the family \cp is in a suitable sense \emph{generic}. We construct a database $\cal D$, supported by an algebraic computation tree, such that for each $x\in [0,1]^n$ the query for the signs of $P_1(x),...,P_s(x)$ can be answered using h d^{\cO(n^2)} comparisons and $nL$ arithmetic operations between real numbers. The arithmetic-geometric tools developed for the construction of $\cal D$ are then employed to exhibit example classes of systems of $n$ polynomial equations in $n$ unknowns whose consistency may be checked using only few arithmetic operations, admitting however an exponential number of comparisons

Polynomial Time Nondimensionalisation of Ordinary Differential Equations via their Lie Point Symmetries

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and translations that are Lie point symmetries of considered ordinary differential system. By rewriting original problem in an invariant coordinates set for these symmetries, one can reduce the involved number of parameters. This process is classically call nondimensionalisation in dimensional analysis. We present an algorithm based on this standpoint and show that its arithmetic complexity is polynomial in input's size

Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case

In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm of \cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem--adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in non-uniform sequential time that is {\em polynomial} in the length of the input (given in straight--line program representation) and an adequately defined {\em geometric degree of the equation system}. Replacing the notion of geometric degree of the given polynomial equation system by a suitably defined {\em real (or complex) degree} of certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.Comment: Late

A probabilistic algorithm to test local algebraic observability in polynomial time

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the \emph{local algebraic observability} problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a \emph{probabilistic seminumerical} algorithm that proposes a solution to this problem in \emph{polynomial time}. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the \emph{complexity of evaluation} of the model and in the number of variables. Furthermore, we show that the \emph{size} of the integers involved in the computations is polynomial in the number of variables and in the degree of the differential system. Last, we estimate the probability of success of our algorithm and we present some benchmarks from our Maple implementation.Comment: 26 pages. A Maple implementation is availabl