Use of algebraically independent numbers for zero recognition of polynomial terms

AbstractA polynomial term is a tree with operators ∗,+,− on the interior nodes and natural numbers and variables on the frontier.We attempt to decide whether or not such a tree represents the zero polynomial by substituting algebraically independent real numbers for the variables and attempting to decide whether or not the resulting constant is zero.From this we get a probabilistic zero recognition test which is somewhat more expensive computationally than the usual method of choosing random integers in a large interval and evaluating, but which depends on the ability to choose a random point in the unit cube and to approximate a polynomial at that point.We also state a conjecture about algebraic independence measure which would give us a deterministic test with a uniformly chosen test point. The result is that if a polynomial term has s(T) nodes, then the bit complexity of deterministic zero recognition is bounded by O(s(T)M(s(T)length(T))), where length(T) measures the length of the term T, and M(n) is the bit complexity of multiplication of two n-digit natural numbers

What is a closed-form number?

If a student asks for an antiderivative of exp(x^2), there is a standard reply: the answer is not an elementary function. But if a student asks for a closed-form expression for the real root of x = cos(x), there is no standard reply. We propose a definition of a closed-form expression for a number (as opposed to a *function*) that we hope will become standard. With our definition, the question of whether the root of x = cos(x) has a closed form is, perhaps surprisingly, still open. We show that Schanuel's conjecture in transcendental number theory resolves questions like this, and we also sketch some connections with Tarski's problem of the decidability of the first-order theory of the reals with exponentiation. Many (hopefully accessible) open problems are described.Comment: 11 pages; submitted to Amer. Math. Monthl

An Algebraic Approach to Hough Transforms

The main purpose of this paper is to lay the foundations of a general theory which encompasses the features of the classical Hough transform and extend them to general algebraic objects such as affine schemes. The main motivation comes from problems of detection of special shapes in medical and astronomical images. The classical Hough transform has been used mainly to detect simple curves such as lines and circles. We generalize this notion using reduced Groebner bases of flat families of affine schemes. To this end we introduce and develop the theory of Hough regularity. The theory is highly effective and we give some examples computed with CoCoA