7 research outputs found
The Hilbert Zonotope and a Polynomial Time Algorithm for Universal Grobner Bases
We provide a polynomial time algorithm for computing the universal Gr\"obner
basis of any polynomial ideal having a finite set of common zeros in fixed
number of variables. One ingredient of our algorithm is an effective
construction of the state polyhedron of any member of the Hilbert scheme
Hilb^d_n of n-long d-variate ideals, enabled by introducing the Hilbert
zonotope H^d_n and showing that it simultaneously refines all state polyhedra
of ideals on Hilb^d_n
Minimal average degree aberration and the state polytope for experimental designs
For a particular experimental design, there is interest in finding which
polynomial models can be identified in the usual regression set up. The
algebraic methods based on Groebner bases provide a systematic way of doing
this. The algebraic method does not in general produce all estimable models but
it can be shown that it yields models which have minimal average degree in a
well-defined sense and in both a weighted and unweighted version. This provides
an alternative measure to that based on "aberration" and moreover is applicable
to any experimental design. A simple algorithm is given and bounds are derived
for the criteria, which may be used to give asymptotic Nyquist-like
estimability rates as model and sample sizes increase
Properties of triangular partitions and their generalizations
Una partició entera es diu triangular si el seu diagrama de Ferrers es pot separar del seu complement amb una línia recta. Aquest treball es basa en alguns desenvolupaments recents sobre el tema per obtenir nous resultats que ofereixen una visió integral de les particions triangulars des de la perspectiva de la combinatòria enumerativa i geomètrica. Es presenta una caracterització natural per a les particions triangulars, la qual es generalitza de forma natural a conceptes anàlegs en dimensions superiors. També es caracteritzen les cel·les esborrables i afegibles, definides per Bergeron i Mazin, i es descriuen els valors de la funció de Möbius al reticle de particions triangulars ordenades per inclusió. Es tracten diversos problemes de compteig, obtenint fórmules d'enumeració i derivant una nova demostració del teorema d'enumeració de paraules equilibrades de Lipatov. Es presenta un algoritme altament eficient per comptar particions triangulars segons la mida, fent possible l'estudi computeritzat de particions triangulars molt grans per primera vegada. La investigació s'estén després a generalitzacions en dimensions superiors anomenades particions piramidals, on es demostra que el nombre de cel·les esborrables i afegibles pot ser arbitràriament gran. En el cas -dimensional per a primer, es descriu el residu del nombre de particions piramidals de la mateixa mida mòdul . Finalment, s'estudien les particions convexes i còncaves, definides com particions amb un diagrama de Ferrers que pot ser separat del seu complement per una corba convexa o còncava. S'obtenen caracteritzacions per a ambdós casos, es demostra que els seus posets segons l'ordre d'inclusió són reticles i es troben els valors de les seves funcions de Möbius. S'estableixen una fita inferior i una superior pel nombre de particions convexes, i s'obté una fita inferior similar en el cas còncau.Una partición entera es triangular si su diagrama de Ferrers puede ser separado de su complemento por una línea recta. Este trabajo se basa en algunos desarrollos recientes sobre el tema con el fin de obtener nuevos resultados que ofrecen una visión integral de las particiones triangulares desde la perspectiva de la combinatoria enumerativa y geométrica. Se presenta una caracterización natural para las particiones triangulares, que se generaliza de manera elegante a conceptos análogos en dimensiones superiores. También se caracterizan las celdas quitables y añadibles, definidas por Bergeron y Mazin, y se describen los valores de la función de Möbius en el retículo de particiones triangulares ordenadas por inclusión. Se abordan diferentes problemas de conteo, obteniendo varias fórmulas de enumeración y derivando una nueva demostración del teorema de enumeración de palabras equilibradas de Lipatov. Se presenta un algoritmo altamente eficiente para contar particiones triangulares por tamaño, lo que permite el estudio computerizado de particiones triangulares grandes por primera vez. La investigación se extiende luego a generalizaciones en dimensiones superiores llamadas particiones piramidales, para las cuales se demuestra que el número de celdas quitables y añadibles puede ser arbitrariamente grande. En el caso -dimensional para primo, se describe el residuo del número de particiones piramidales de un mismo tamaño módulo . Finalmente, se estudian las particiones convexas y cóncavas, definidas como particiones cuyo diagrama de Ferrers puede ser separado de su complemento por una curva convexa o cóncava. Ambos casos se caracterizan, se demuestra que sus posets bajo el orden de inclusión son retículos y se encuentran los valores de sus funciones de Möbius. Se proporciona una cota inferior y superior para el número de particiones convexas y se obtiene una cota inferior similar en el caso cóncavo.An integer partition is said to be triangular if its Ferrers diagram can be separated from its complement by a straight line. This work builds on some recent developments on the topic in order to obtain new results that offer a comprehensive look on triangular partitions from the lens of enumerative and geometric combinatorics. A natural characterization for triangular partitions is given, and it is generalized nicely to higher-dimensional analogues. The removable and addable cells, defined by Bergeron and Mazin, are also characterized, and the values of the Möbius function in the lattice of triangular partitions ordered by containment are described. Different counting problems are tackled, obtaining several enumeration formulas and deriving a new proof of Lipatov's enumeration theorem for balanced words. A highly efficient algorithm to count triangular partitions by size is presented, allowing the computerized study of large triangular partitions for the first time. The research is then extended to higher-dimensional generalizations called pyramidal partitions, for which it is proved that the number of removable and addable cells can be arbitrarily high. In the -dimensional case for prime, the residue of the number of pyramidal partitions of the same size modulo is described. Finally, we study convex and concave partitions, defined as partitions whose Ferrers diagram can be separated from its complement by a convex or concave curve. Both cases are characterized, it is proved that their posets under the containment order are lattices and the values of their Möbius functions are found. A lower and an upper bound are given for the number of convex partitions and a similar lower bound is obtained in the concave case.Outgoin
On the number of corner cuts
A corner cut in dimension d is a finite subset of N d 0 that can be separated from its complement in N d 0 by an affine hyperplane disjoint from N d 0. Corner cuts were first investigated by Onn and Sturmfels [6], the original motivation stemming from computational commutative algebra. Let us write N d 0 k¡cut for the set of corner cuts of cardinality k; in the computational geometer’s termi-nology, these are the k-sets of N d 0. Among other things, Onn and Sturmfels give an upper bound of O(k d−1 2d d+1) for the size of N d 0 k¡cut when the dimension is fixed. In two dimensions, it is known (see [3]) that # N d 0 = Θ(k log k). k¡cut We will see that in general, for any fixed dimension d, the order of magnitude of # N d 0 k¡cut is between kd−1 log k and (k log k) d−1. (It has been communicated to me that the same bounds have been found independently by Gaël Rémond.) In fact, the elements of N d 0 k¡cut correspond to the vertices of a certain polytope, and what our proof will show is that the above upper bound holds for the total number of flags of that polytope. Key words: corner cut, Gröbner fan, k-set, reverse search.