The Casas-Alvero conjecture for infinitely many degrees

Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been some computational verifications for small degrees d. In this paper the conjecture is proved in the case where the degree of the polynomial is a power of a prime number, or twice such a power. Moreover, for each positive characteristic p, we give an example of a polynomial of degree d which is not a dth power but which has a common factor with each of its first d-1 derivatives. This shows that the assumption of characteristic zero is essential for the converse statement to hold.Comment: 7 pages; v2: corrected some typos and references, and added section on computational aspect

Non-ruled surfaces with straight lines for architecture = Superfícies não regradas com rectas para a Arquitectura

In architecture, ruled surfaces are well-known. It is less known that there are also many mathematical surfaces which contain just a small number of straight lines. In this talk I will discuss some opportunities about using those in architecture from a mathematician’s point of view. The examples which we will present have a wide variety ranging from classical mathematical surfaces (which may only be used exactly as they are) to modern surfaces which may be designed in a collaboration between mathematicians and architects in order to suit the needs of a particular architectural project. Na arquitectura, as superfícies regradas são bem conhecidas. É menos conhecido o facto de haver muitas superfícies matemáticas que contêm apenas algumas rectas. Nesta conversa discutirei algumas das oportunidades da utilização dessas na Arquitectura de um ponto de vista matemático. O exemplo que se apresentará tem um intervalo de variedade muito amplo desde as superfícies matemáticas clássicas (que só podem ser utilizadas tal como são) até às superfícies modernas que podem ser desenhadas numa colaboração entre matemáticos e arquitectos de maneira a servir as necessidades específicas de um projecto em particular

Hypersurfaces with many singularities

1. Teil: Bekannte Konstruktionen. Die vorliegende Arbeit gibt zunächst einen ausführlichen Überblick über die bisherigen Entwicklungen auf dem klassischen Gebiet der Hyperflächen mit vielen Singularitäten. Die maximale Anzahl mu^n(d) von Singularitäten auf einer Hyperfläche vom Grad d im P^n(C) ist nur in sehr wenigen Fällen bekannt, im P^3(C) beispielsweise nur für d<=6. Abgesehen von solchen Ausnahmen existieren nur obere und untere Schranken. 2. Teil: Neue Konstruktionen. Für kleine Grade d ist es oft möglich, bessere Resultate zu erhalten als jene, die durch allgemeine Schranken gegeben sind. In dieser Arbeit beschreiben wir einige algorithmische Ansätze hierfür, von denen einer Computer Algebra in Charakteristik 0 benutzt. Unsere anderen algorithmischen Methoden basieren auf einer Suche über endlichen Körpern. Das Liften der so experimentell gefundenen Hyperflächen durch Ausnutzung ihrer Geometrie oder Arithmetik liefert beispielsweise eine Fläche vom Grad 7 mit $99$ reellen gewöhnlichen Doppelpunkten und eine Fläche vom Grad 9 mit 226 gewöhnlichen Doppelpunkten. Diese Konstruktionen liefern die ersten unteren Schranken für mu^3(d) für ungeraden Grad d>5, die die allgemeine Schranke übertreffen. Unser Algorithmus hat außerdem das Potential, auf viele weitere Probleme der algebraischen Geometrie angewendet zu werden. Neben diesen algorithmischen Methoden beschreiben wir eine Konstruktion von Hyperflächen vom Grad d im P^n mit vielen A_j-Singularitäten, j>=2. Diese Beispiele, deren Existenz wir mit Hilfe der Theorie der Dessins d'Enfants beweisen, übertreffen die bekannten unteren Schranken in den meisten Fällen und ergeben insbesondere neue asymptotische untere Schranken für j>=2, n>=3. 3. Teil: Visualisierung. Wir beschließen unsere Arbeit mit einer Anwendung unserer neuen Visualisierungs-Software surfex, die die Stärken mehrerer existierender Programme bündelt, auf die Konstruktion affiner Gleichungen aller 45 topologischen Typen reeller kubischer Flächen.Part 1: Known Constructions. The present work starts with a large survey of the classical subject of hypersurfaces with many singularities. The maximum number mu^n(d) of singularities on a hypersurface of degree d in P^n(C) is known in very few cases only, e.g. in P^3(C) for d<=6. Apart from such exceptions, there only exist upper and lower bounds. Part 2: New Constructions. For low degree d, it is often possible to obtain better results than those given by the general bounds. In this work, we present some algorithmic approaches for these cases, one of which uses computer algebra in characteristic 0. The other algorithmic methods are based on a search over finite fields. Lifting the experimentally found hypersurfaces using their geometry or arithmetic yields e.g. a surface of degree 7 with 99 real ordinary double points and a surface of degree 9 with 226 ordinary double points. These constructions give the first lower bounds for mu^3(d) for odd degree d>5 which exceed the general bound. Moreover, our algorithm has the potential of being applied to many other concrete problems in algebraic geometry. Besides these algorithmic approaches, we describe a construction of hypersurfaces of degree d in P^n with many A_j-singularities, j>=2. The proof of their existence is based on the theory of dessins d'enfants. These constructions exceed the known lower bounds in most cases and yield in particular new asymptotic lower bounds for all j>=2, n>=3. Part 3: Visualization. We close this work with an application of our new visualization software surfex which combines the strengths of several existing programs: We describe how to construct nice affine equations for all 45 topological types of real cubic surfaces

Surfaces with many solitary points

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