# Some algebraic and geometric computations in PSPACE

@inproceedings{Canny1988SomeAA, title={Some algebraic and geometric computations in PSPACE}, author={John F. Canny}, booktitle={Symposium on the Theory of Computing}, year={1988} }

We give a PSPACE algorithm for determining the signs of multivariate polynomials at the common zeros of a system of polynomial equations. One of the consequences of this result is that the “Generalized Movers' Problem” in robotics drops from EXPTIME into PSPACE, and is therefore PSPACE-complete by a previous hardness result [Rei]. We also show that the existential theory of the real numbers can be decided in PSPACE. Other geometric problems that also drop into PSPACE include the 3-d Euclidean…

## 643 Citations

### The Hardness of Polynomial Equation Solving

- Mathematics, Computer ScienceFound. Comput. Math.
- 2003

This paper investigates the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory and admitting the representation of certain limit objects.

### A computational proof of complexity of some restricted counting problems

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2009

### A Gröbner Free Alternative for Polynomial System Solving

- Mathematics, Computer ScienceJ. Complex.
- 2001

A new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials are introduced, and a new codification of the set of solutions of a positive dimensional algebraic variety is given relying on a new global version of Newton's iterator.

### A PSPACE construction of a hitting set for the closure of small algebraic circuits

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2017

It is shown that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known.

### On the Complexity of Counting Irreducible Components and Computing Betti Numbers of Algebraic Varieties

- Mathematics, Computer Science
- 2007

This thesis gives a uniform method for the two problems #CCC and #ICC of counting the connected and irreducible components of complex algebraic varieties and proves that the problem of deciding connectedness of a complex affine or projective variety given over the rationals is PSPACE-hard.

### On the combinatorial and algebraic complexity of quantifier elimination

- Mathematics, Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

An improved bound on the radius of a ball centered at the origin, which is guaranteed to intersect every connected component of the sign partition induced by a family of polynomials is given.

### On the Combinatorial and Algebraic Complexity of Quantifier Elimination

- Mathematics, Computer ScienceFOCS
- 1994

An improved bound on the radius of a ball centered at the origin, which is guaranteed to intersect every connected component of the sign partition induced by a family of polynomials is given.

### Algebraic Geometry Over Four Rings and the Frontier to Tractability

- Mathematics
- 2000

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of…

### Differential forms in computational algebraic geometry

- Computer Science, MathematicsISSAC '07
- 2007

A randomised algorithm solving #ICC for a fixed number of rational equations given by straight-line programs (slps), which runs in parallel polylogarithmic time in the length and the degree of the slps.

### First Steps in Algorithmic Fewnomial Theory

- Mathematics
- 2004

A phase-transition is described for when m is large enough to make FEAS_R be NP-hard, when restricted to inputs consisting of a single n-variate polynomial with exactly m monomial terms:Polynomial-time for m 0).

## References

SHOWING 1-10 OF 18 REFERENCES

### On Euclid's Algorithm and the Theory of Subresultants

- MathematicsJACM
- 1971

An elementary treatment of the theory of subresultants is presented, and the relationship of the sub resultants of a given pair of polynomials to their polynomial remainder sequence as determined by Euclid's algorithm is examined.

### Fast parallel matrix inversion algorithms

- Computer Science, Mathematics16th Annual Symposium on Foundations of Computer Science (sfcs 1975)
- 1975

It will be shown in the sequel that the parallel arithmetic complexity of all these four problems is upper bounded by O(log2n) and the algorithms that establish this bound use a number of processors polynomial in n, disproves I. Munro's conjecture.

### The complexity of robot motion planning

- Computer Science
- 1988

John Canny resolves long-standing problems concerning the complexity of motion planning and, for the central problem of finding a collision free path for a jointed robot in the presence of obstacles, obtains exponential speedups over existing algorithms by applying high-powered new mathematical techniques.

### Computer algebra symbolic and algebraic computation

- Mathematics, Computer ScienceSIGS
- 1982

This volume is the first systematic and complete treatment of computergebra and presents the basic problems of computer algebra and the best algorithms now known for their solution with their mathematical foundations, and complete references to the original literature.

### On shortest paths in polyhedral spaces

- MathematicsSTOC '84
- 1984

The main result of this paper involves a favorable special case of the 3-D shortest path problem, namely that of finding the shortest path between two points along the surface of a convex polyhedron.

### On the Worst-Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials

- Mathematics, Computer ScienceSIAM J. Comput.
- 1989

It is shown that with respect to a certain model of computation, the worst-case computational complexity of obtaining $\varepsilon $-approximations to at least those zeros $\xi $ satisfying $|\xi | \leqq R$ for arbitrary $f \in \mathcal{P}$ is $\Theta (\log \log (({ R / \varpsilon })$; that is to say, both upper and lower bounds are proved.

### Shortest Paths in Euclidean Space with Polyhedral Obstacles.

- Mathematics
- 1985

Abstract : This document considers the problem of finding a minimum length path between two points in Euclidean space which avoids a set (not necessarily convex) polyhedral obstacles; we let n denote…

### Some Useful Bounds

- Mathematics
- 1983

Some fundamental inequalities for the following values are listed: the determinant of a matrix, the absolute value of the roots of a polynomial, the coefficients of divisors of polynomials, and the…

### Topological Stability of Smooth Mappings

- Mathematics
- 1976

Construction of canonical stratifications.- Stratifications and flows.- Unfoldings of smooth map-germs.- Proof of the topological stability theorem.