Polynomial Systems

The problem is to identify a movable object that is in some sense known, if it is encountered later. Suppose we have a sensor, on a fixed radar station or a moving platform. We have an object, say object A, previously measured, with certain distinct identifiable points p i. We know the distances between these points. We later encounter a similar object B and want to know if it is A. We have a sensor that sends and receives electronic signals, and so we measure the distances t i from the sensor to the distinguished points on B. We first consider the two-dimensional case. Assume there are three distinct points on A. We have our measured distances t 1 , t 2 , t 3 and previously known distances between the points on A, d 1 , d 2 , d 3. We derive a polynomial system relating these quantities and show that it is easy to solve yielding a resultant that is the "signature" for A. Its use will eliminate B if B is not A. The generalization to three dimensions is immediate. We need a fourth point. The polynomial system contains many parameters, but we solve it symbolically. We then discuss generalizations involving flexibility. In those cases we need five points and the systems are much more complex. In section five we consider the same question of recognition but using a different type of sensor, one that can only receive. We compare a Gröbner basis approach with the Dixon resultant method. We compare solutions on Magma, Maple, and Fermat computer algebra systems .

In this work "solve a polynomial system" means to search
for the common roots of a set of multivariate polynomials. Usually there
are n variables, n equations, and some parameters. The system is neither
over- nor under-determined. We eliminate all but one of the variables,
leaving one polynomial in one variable and the parameters { the resultant
[2].
The Bezout-Dixon method produces a matrix, M; whose determinant
Det[M] is a multiple of the resultant. Dixon-EDF [7, 9] is a way to com-
pute the resultant without nding the entire determinant.
In this work we present four new signicant acceleration techniques and
extensions to EDF.
- Ordering of variables. We give a heuristic for the "weight" of a vari-
able within the system. The variables should be given precedence using
this order, with the heaviest having the highest precedence.
- Leaving M as a 2  2 matrix. Dixon-EDF normalizes M in a special
way. When nished, M is the identity matrix. But there are dicult
problems where by the 2  2 step, the four polynomials are too large to
multiply. Simply leaving M in that state can be acceptable.
- Decomposing into blocks. It has long been noted that Det[M] is
often of the form qrk, where q is of no interest and r is the resultant.
This suggests that matrix M could be decomposed into k equivalent
blocks. We present a fast way to produce this decomposition, if present,
and show huge speed-ups are possible.
- Dealing with more equations than variables. If there are more
equations than variables, the Dixon resultant cannot be used even if
the solution set is zero-dimensional. We present a method that can be
eective in converting the system into one that Dixon can solve.
Each new method will be illustrated with examples showing its eective-
ness.

A fundamental problem in computer graphics and computer aided design is to convert between a parameterization of a surface and an implicit representation of it. Almost as fundamental is to derive a parameterization for the intersection of two surfaces. In these problems, it seems that resultants, specifically the Dixon resultant , have been underappreciated. Indeed, several well known papers from ten to twenty years ago reported unsuitability of resultant techniques. To the contrary, we show that the Dixon resultant is an extremely effective and efficient method to compute an implicit representation. To use resultants to compute a parameterization of an intersection, we introduce the concept of an "implicit parameterization." Unlike the conventional parameterization of a curve where x, y, and z are each explicitly given as functions of, say, t, we have three implicit functions, one each for (x, t), (y, t), and (z, t). This concept has rarely been mentioned before. We show that given a (conventional) parameterization for one surface and either an implicit equation for the second, or a parameterization for it, it is straightforward to compute an implicit parameterization for the intersection. Doing so is very easy for the Dixon resultant, but can be very daunting even for well respected Gröbner bases programs. Further, we demonstrate that such implicit parameterizations are useful. We use builtin 3D plotting utilities of a computer algebra system to graph the intersection using our implicit parameterization. We do this for examples that are more complex than the quadric examples usually discussed in intersection papers.

Most beta scientists know what polynomials are how they behave and what to do with them. But how about polynomials with broken or fractional powers. This is a small study to what they are and how they behave. In other words what if the parameters p, q, r, s are not whole numbers in Y = a + bX p +cX q + dX r + eX s + .... but fractions like 0.95, 1.13, 2.25, 3.025 etc.

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R there exists a radially symmetric stationary solution with free boundary r = R and interior boundary r = ρ.The system depends on a positive parameter μ, and for a sequence of values μ2 < μ3 < · · · there also exist branches of symmetric breaking stationary solutions, which bifurcate from these values. We use a homotopy method on the polynomial system associated to the discretization of the free boundary problem to compute the nonradial symmetric solutions and discuss their linear stability.