scholarly journals On multivariate polynomials with many roots over a finite grid

2020 ◽  
pp. 2150136
Author(s):  
Olav Geil
Keyword(s):  
Large Class ◽  
Upper Bound ◽  
Multivariate Polynomials ◽  
Irreducible Polynomials ◽  
Monomial Ordering ◽  
The One ◽  
Bézout’S Theorem

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout’s theorem, IEEE Trans. Inform. Theory 46(2) (2000) 635–641, On (or in) Dick Blahut’s ‘footprint’, Codes[Formula: see text] Curves Signals (1998) 3–9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

ON CONDENSATION IN NON-IDEAL BOSE GAS

1989 ◽  
Vol 03 (06) ◽  
pp. 471-478
Author(s):  
D.P. SANKOVICH
Keyword(s):  
Critical Temperature ◽  
Low Temperature ◽  
Upper Bound ◽  
Bose Gas ◽  
The One ◽  
Ideal Bose Gas

A model of the non-ideal Bose gas is considered. We prove the existence of condensate in the model at sufficiently low temperature. The method of majorizing estimates for the Duhamel Two Point Functions is used. The equation for the critical temperature and the upper bound for the one-particle excitations energy are obtained.

On the braid index of links with nested diagrams

1992 ◽  
Vol 111 (2) ◽  
pp. 273-281 ◽  
Author(s):  
D. A. Chalcraft
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Upper Bound ◽  
Simple Criterion ◽  
Braid Index

AbstractThe number of Seifert circuits in a diagram of a link is well known 9 to be an upper bound for the braid index of the link. The -breadth of the so-called P-polynomial 3 of the link is known 5, 2 to give a lower bound. In this paper we consider a large class of links diagrams, including all diagrams where the interior of every Seifert circuit is empty. We show that either these bounds coincide, or else the upper bound is not sharp, and we obtain a very simple criterion for distinguishing these cases.

scholarly journals Planet formation by pebble accretion in ringed disks

2020 ◽  
Vol 638 ◽  
pp. A1 ◽  
Author(s):  
A. Morbidelli
Keyword(s):  
Growth Rate ◽  
Upper Bound ◽  
Planet Formation ◽  
Accretion Rate ◽  
Central Star ◽  
Giant Planets ◽  
Type I ◽  
Dust Density ◽  
Dominant Process ◽  
The One

Context. Pebble accretion is expected to be the dominant process for the formation of massive solid planets, such as the cores of giant planets and super-Earths. So far, this process has been studied under the assumption that dust coagulates and drifts throughout the full protoplanetary disk. However, observations show that many disks are structured in rings that may be due to pressure maxima, preventing the global radial drift of the dust. Aims. We aim to study how the pebble-accretion paradigm changes if the dust is confined in a ring. Methods. Our approach is mostly analytic. We derived a formula that provides an upper bound to the growth of a planet as a function of time. We also numerically implemented the analytic formulæ to compute the growth of a planet located in a typical ring observed in the DSHARP survey, as well as in a putative ring rescaled at 5 AU. Results. Planet Type I migration is stopped in a ring, but not necessarily at its center. If the entropy-driven corotation torque is desaturated, the planet is located in a region with low dust density, which severely limits its accretion rate. If the planet is instead near the ring’s center, its accretion rate can be similar to the one it would have in a classic (ringless) disk of equivalent dust density. However, the growth rate of the planet is limited by the diffusion of dust in the ring, and the final planet mass is bounded by the total ring mass. The DSHARP rings are too far from the star to allow the formation of massive planets within the disk’s lifetime. However, a similar ring rescaled to 5 AU could lead to the formation of a planet incorporating the full ring mass in less than 1/2 My. Conclusions. The existence of rings may not be an obstacle to planet formation by pebble-accretion. However, for accretion to be effective, the resting position of the planet has to be relatively near the ring’s center, and the ring needs to be not too far from the central star. The formation of planets in rings can explain the existence of giant planets with core masses smaller than the so-called pebble isolation mass.

Church-Rosser theorem for typed functional systems

1985 ◽  
Vol 50 (3) ◽  
pp. 782-790 ◽  
Author(s):  
George Koletsos
Keyword(s):  
Upper Bound ◽  
Type Structure ◽  
Functional Systems ◽  
Normalization Theorem ◽  
One Step ◽  
The One ◽  
Reduction Relation ◽  
The Church ◽  
The Way

Introduction. This paper contains a new proof of the Church-Rosser theorem for the typed λ-calculus, which also applies to systems with infinitely long terms.The ordinary proof of the Church-Rosser theorem for the general untyped calculus goes as follows (see [1]). If is the binary reduction relation between the terms we define the one-step reduction 1 in such a way that the following lemma is valid.Lemma. For all terms a and b we have: ab if and only if there is a sequence a = a0, …, an = b, n ≥ 0, such that aiiai + 1for 0 ≤ i < n.We then prove the Church-Rosser property for the relation 1 by induction on the length of the reductions. And by combining this result with the above lemma we obtain the Church-Rosser theorem for the relation .Unfortunately when we come to infinite terms the above lemma is not valid anymore. The difficulty is that, assuming the hypothesis for the infinitely many premises of the infinite rule, there may not exist an upper bound for the lengths n of the sequences ai = a0, …, an = bi (i < α); cf. the infinite rule (iv) in §6.A completely new idea in the case of the typed λ-calculus would be to exploit the type structure in the way Tait did in order to prove the normalization theorem. In this we succeed by defining a suitable predicate, the monovaluedness predicate, defined over the type structure and having some nice properties. The key notion permitting to define this predicate is the notion of I-form term (see below). This Tait-type proof has a merit, namely that it can be extended immediately to the case of infinite terms.

scholarly journals ON THE CHACTERISTIC NUMBERS OF VOTING GAMES

2006 ◽  
Vol 08 (04) ◽  
pp. 643-654 ◽  
Author(s):  
MATHIEU MARTIN ◽  
VINCENT MERLIN
Keyword(s):  
Upper Bound ◽  
Solution Concept ◽  
Voting Games ◽  
Voting Game ◽  
The Stability ◽  
The One

This paper deals with the non-emptiness of the stability set for any proper voting game. We present an upper bound on the number of alternatives which guarantees the non emptiness of this solution concept. We show that this bound is greater than or equal to the one given by Le Breton and Salles (1990) for quota games.

scholarly journals Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

2018 ◽  
Vol 28 (2) ◽  
pp. 253-279
Author(s):  
O. GEIL ◽  
U. MARTÍNEZ-PEÑAS
Keyword(s):  
Upper Bound ◽  
Existence And Uniqueness ◽  
Combinatorial Nullstellensatz ◽  
Finite Collection ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Evaluation Codes ◽  
Number Of Zeros ◽  
Interpolating Polynomials ◽  
Common Zeros

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

scholarly journals SKEW CATEGORY ALGEBRAS ASSOCIATED WITH PARTIALLY DEFINED DYNAMICAL SYSTEMS

2012 ◽  
Vol 23 (04) ◽  
pp. 1250040 ◽  
Author(s):  
PATRIK LUNDSTRÖM ◽  
JOHAN ÖINERT
Keyword(s):  
Dynamical Systems ◽  
Topological Space ◽  
Large Class ◽  
Additive Group ◽  
Nonzero Ideal ◽  
Intersection Property ◽  
The Other ◽  
The One ◽  
Category Algebras ◽  
The Ideal

We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Topop and show that it defines what we call a skew category algebra A ⋊σ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊σ G and, on the other hand, maximal commutativity of A in A ⋊σ G. In particular, we show that if G is a groupoid and for each e ∈ ob (G) the group of all morphisms e → e is countable and the topological space s(e) is Tychonoff and Baire. Then the following assertions are equivalent: (i) s is topologically free; (ii) A has the ideal intersection property, i.e. if I is a nonzero ideal of A ⋊σ G, then I ∩ A ≠ {0}; (iii) the ring A is a maximal abelian complex subalgebra of A ⋊σ G. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.

scholarly journals Subnomality in soluble minimax groups

1974 ◽  
Vol 17 (1) ◽  
pp. 113-128 ◽  
Author(s):  
D. J. McCaughan
Keyword(s):  
Large Class ◽  
Upper Bound ◽  
Nilpotent Groups ◽  
Subnormal Subgroup ◽  
Wreath Products ◽  
Minimal Length ◽  
Intersection Property ◽  
Finite Chain ◽  
Subnormal Subgroups

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.

scholarly journals A SHARP UPPER BOUND FOR REGION UNKNOTTING NUMBER OF TORUS KNOTS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350019 ◽  
Author(s):  
SIWACH VIKASH ◽  
MADETI PRABHAKAR
Keyword(s):  
Large Class ◽  
Upper Bound ◽  
Torus Knots ◽  
Torus Links

Region crossing change for a knot or a proper link is an unknotting operation. In this paper, we provide a sharp upper bound on the region unknotting number for a large class of torus knots and proper links. Also, we discuss conditions on torus links to be proper.

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