15 research outputs found
An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance
Cylindrical algebraic decomposition (CAD) is an important tool for the
investigation of semi-algebraic sets, with applications within algebraic
geometry and beyond. We recently reported on a new implementation of CAD in
Maple which implemented the original algorithm of Collins and the subsequent
improvement to projection by McCallum. Our implementation was in contrast to
Maple's in-built CAD command, based on a quite separate theory. Although
initially developed as an investigative tool to compare the algorithms, we
found and reported that our code offered functionality not currently available
in any other existing implementations. One particularly important piece of
functionality is the ability to produce order-invariant CADs. This has allowed
us to extend the implementation to produce CADs invariant with respect to
either equational constraints (ECCADs) or the truth-tables of sequences of
formulae (TTICADs). This new functionality is contained in the second release
of our code, along with commands to consider problem formulation which can be a
major factor in the tractability of a CAD. In the report we describe the new
functionality and some theoretical discoveries it prompted. We describe how the
CADs produced using equational constraints are able to take advantage of not
just improved projection but also improvements in the lifting phase. We also
present an extension to the original TTICAD algorithm which increases both the
applicability of TTICAD and its relative benefit over other algorithms. The
code and an introductory Maple worksheet / pdf demonstrating the full
functionality of the package are freely available online.Comment: 12 pages; University of Bath, Dept. Computer Science Technical Report
Series, 2013-02, 201
Constructing Fewer Open Cells by GCD Computation in CAD Projection
A new projection operator based on cylindrical algebraic decomposition (CAD)
is proposed. The new operator computes the intersection of projection factor
sets produced by different CAD projection orders. In other words, it computes
the gcd of projection polynomials in the same variables produced by different
CAD projection orders. We prove that the new operator still guarantees
obtaining at least one sample point from every connected component of the
highest dimension, and therefore, can be used for testing semi-definiteness of
polynomials. Although the complexity of the new method is still doubly
exponential, in many cases, the new operator does produce smaller projection
factor sets and fewer open cells. Some examples of testing semi-definiteness of
polynomials, which are difficult to be solved by existing tools, have been
worked out efficiently by our program based on the new method.Comment: Accepted by ISSAC 2014 (July 23--25, 2014, Kobe, Japan
Finding best possible constant for a polynomial inequality
Given a multi-variant polynomial inequality with a parameter, how to find the best possible value of this parameter that satisfies the inequality? For instance, find the greatest number k that satisfies a 3 +b 3 +c 3 +k(a 2 b+b 2 c+c 2 a)−(k+1)(ab 2 +bc 2 +ca 2 ) ≥ 0 for all nonnegative real numbers a, b, c. Analogues problems often appeared in studies of inequalities and were dealt with by various methods. In this paper, a general algorithm is proposed for finding the required best possible constant. The algorithm can be easily implemented by computer algebra tools such as Maple
Validity proof of Lazard's method for CAD construction
In 1994 Lazard proposed an improved method for cylindrical algebraic
decomposition (CAD). The method comprised a simplified projection operation
together with a generalized cell lifting (that is, stack construction)
technique. For the proof of the method's validity Lazard introduced a new
notion of valuation of a multivariate polynomial at a point. However a gap in
one of the key supporting results for his proof was subsequently noticed. In
the present paper we provide a complete validity proof of Lazard's method. Our
proof is based on the classical parametrized version of Puiseux's theorem and
basic properties of Lazard's valuation. This result is significant because
Lazard's method can be applied to any finite family of polynomials, without any
assumption on the system of coordinates. It therefore has wider applicability
and may be more efficient than other projection and lifting schemes for CAD.Comment: 21 page