Abstract. A current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper contributes to this by defining a new criterion for assessing path-following tasks in the planar case, that is, evaluating the performance of robots that are required to follow a desired reference path. Such criterion comes from the study of the local differential geometry of the problem. New conditions for deciding whether or not the zero locus of a given polynomial intersects the neighbourhood of a point are defined. Based on this, new algorithms are presented and tested on both simulated data and experiments conducted at sea employing an Unmanned Surface Vehicle.2000 Mathematics Subject Classifications: 26C10, 15A60, 93C85, 62P99.

A complete intersection of n polynomials in n indeterminates has only a finite number of zeros. In this paper we address the following question: how do the zeros change when the coefficients of the polynomials are perturbed? In the first part we show how to construct semi-algebraic sets in the parameter space over which all the complete intersection ideals share the same number of isolated real zeros. In the second part we show how to modify the complete intersection and get a new one which generates the same ideal but whose real zeros are more stable with respect to perturbations of the coefficients.

In 1884 the German mathematician Karl Rohn published a substantial paper on [11] on the properties of quartic surfaces with triple points, proving (among many other things) that the maximum number of lines contained in a quartic monoid surface is 31.In this paper we study in details this class of surfaces. We prove that there exists an open subset A ⊆ P 1 K (K is a characteristic zero field) that parametrizes (up to a projectivity) all the quartic monoid surfaces with 31 lines; then we study the action of PGL(4, K) on these surfaces, we show that the stabiliser of each of them is a group isomorphic to S 3 except for one surface of the family, whose stabiliser is a group isomorphic to S 3 × C 3 . Finally we show that the j-invariant allows one to decide, also in this situation, when two elements of A give the same surface up to a projectivity.To get our results, several computational tools, available in computer algebra systems, are used.

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