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### The partition polynomial of a finite set system

David G Wagner
1991 Journal of combinatorial theory. Series A
We introduce the partition polynomial of a finite set system, which generalizes the matching polynomial of a graph, and elucidate some of its properties.  ...  The properties of the partition polynomial with respect to composition of set systems also prove interesting; the main result is an extension of the Heilmann-Lieb Theorem to this context. 0 1991 Academic  ...  We present here a third level of generality, by defining a partition polynomial for any finite set system.  ...

### Physical Computation, P/poly and P/log*

Richard Whyman
2016 Electronic Proceedings in Theoretical Computer Science
Using this we describe a class of classical "physical" computation systems whose computational capabilities in polynomial time are equivalent to P/poly.  ...  Finally we describe two classes of classical "physical" computation systems in this new framework whose computational capabilities in polynomial time are equivalent to P/poly and P/log*.  ...  A computation system is a quadruple C = (X , Π, T , x 0 ) where: • X is a non-empty set, • Π is a finite non-empty set of finite partitions 4 of X , • T is a finite non-empty set of transformations T :  ...

### Page 11 of Mathematical Reviews Vol. 38, Issue 1 [page]

1969 Mathematical Reviews
Kleitman, D. 56 On families of subsets of a finite set containing no two disjoint sets and their union. J. Combinatorial Theory 5 (1968), 235-237. Let S be a finite set, k an integer, and |S|=3k+1.  ...  By a partition poly- nomial is understood a multivariable polynomial defined by a sum over partitions of its index.  ...

### Split absolutely irreducible integer-valued polynomials over discrete valuation domains [article]

Sophie Frisch, Sarah Nakato, Roswitha Rissner
2022 arXiv   pre-print
For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors times  ...  They correspond bijectively to finite sets, which we call balanced, characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of M.  ...  the remaining elements of S, together with S as a system of representatives, constitute a pointed M -adic partition of R.  ...

### Proving Geometric Theorems by Partitioned-Parametric Gröbner Bases [chapter]

Xuefeng Chen, Peng Li, Long Lin, Dingkang Wang
2006 Lecture Notes in Computer Science
The notion of partitioned-parametric Gröbner bases of a polynomial ideal under constraints is introduced and an algorithm for constructing partitioned-parametric Gröbner bases is given; the correctness  ...  By this method, besides proving the generic truth of a geometric theorem, we can give the necessary and sufficient conditions on the free parameters for the theorem to be true.  ...  Since f has a finite number of terms, the above process will terminate in finite steps and the number of the unambiguous polynomial sets in the parametric partition of (C, {f }) is also finite.  ...

### Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials [article]

Thi Xuan Vu
2022 arXiv   pre-print
Given a sequence of polynomials 𝐟 = (f_1, ..., f_s) and a polynomial ϕ in 𝕂[x_1, ..., x_n] with s<n, we consider the problem of computing the set W(ϕ, 𝐟) of points at which 𝐟 vanishes and the Jacobian  ...  The runtime of our algorithm is polynomial in the total number of points described by the output.  ...  We define K[ ] := { ∈ K[ ] : ( ) = for all ∈ } the set of all polynomial functions invariant by the action of . Let be a finite group.  ...

### PARTITION FUNCTION ZEROS OF AN ISING SPIN GLASS

P. H. DAMGAARD, J. LACKI
1996 International Journal of Modern Physics C
In any finite volume, the simultaneous distribution of the zeros of all partition functions can be viewed as part of the more general problem of finding the location of all the zeros of a certain class  ...  We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes.  ...  These authors demonstrate that a spin model defined on a regular lattice can lead to a fractal distribution of partition function zeros.  ...

### Page 630 of Mathematical Reviews Vol. , Issue 92b [page]

1992 Mathematical Reviews
92b:05005 92b:05005 05A18 05A15 05C70 Wagner, David G. (1-MIT) The partition polynomial of a finite set system. J. Combin. Theory Ser. A 56 (1991), no. 1, 138-159.  ...  This polynomial is the rank-generating function of set partitions whose blocks come from a given set system (collection of subsets).  ...

### Dynamics of piecewise isometries

Arek Goetz
2000 Illinois Journal of Mathematics
We begin a systematic study of Euclidean piecewise isometric dynamical systems (p.i.d.s.) with a particular focus on the interplay between geometry, symbolic dynamics, and the group of isometries associated  ...  This theoretical investigation is motivated by the many examples of piecewise isometric dynamical systems found recently in the literature.  ...  A number of final suggestions were made by Sheldon Axler.  ...

### Page 1826 of Automation and Remote Control Vol. 54, Issue 12 [page]

1993 Automation and Remote Control
Let us partition the set J] = {1,..., m} into three non-intersecting sets: I=AUBUC (some of A, B, C may be empty).  ...  Consequently, the number of one-parametric families of regular polynomials is finite.  ...

### Quasi-polynomials, linear Diophantine equations and semi-linear sets

Flavio D'Alessandro, Benedetto Intrigila, Stefano Varricchio
2012 Theoretical Computer Science
We study the growth function of semi-linear sets and we prove that such a function is a piecewise quasi-polynomial on a polyhedral partition of N t .  ...  Moreover, we give a new proof of combinatorial character of a famous theorem by Dahmen and Micchelli on the partition function of a system of Diophantine linear equations.  ...  We also thank the anonymous referee for her (his) help in greatly improving a previous version of this paper.  ...

### Page 4179 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews
In order to guarantee that this principle leads to the complete factorization of any such poly- nomial f(x), one has to work with a pair-splitting set of partitions, i.e., a set of partitions such that  ...  The au- thors propose to use pair-splitting sets of partitions obtained from parallel classes of k-flats in a suitable affine geometry.  ...

### Solution Regularity of k-partite Linear Systems – A Variant of Rado's Theorem [article]

Hongyi Zhou
2021 arXiv   pre-print
The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to have a monochromatic solution whenever the positive integers are finitely colored.  ...  For k ≥ 2, we give some necessary and sufficient conditions such that, when the set of variables is partitioned into k subsets, there is a solution such that the variables of each subset are monochromatic  ...  To help describe the partitions, we say an r-coloring of elements in a set S is a function χ : S Ñ C, where C is the set of colors.  ...

### Page 5823 of Mathematical Reviews Vol. , Issue 96j [page]

1996 Mathematical Reviews
Examples are also given of a finite group without a finite basis of collective identities and a completely 0- simple semigroup with a finite basis of identities but without a finite system of disjunctive  ...  .], in his studies on the law of quadratic reciprocity, used a partitioning of the set of integral binary quadratic forms with a given dis- criminant into classes which he called genera.  ...

### New polynomial and multidimensional extensions of classical partition results [article]

Vitaly Bergelson, John H. Johnson Jr., Joel Moreira
2016 arXiv   pre-print
We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of (m,p,c)-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding  ...  In the 1970s Deuber introduced the notion of (m,p,c)-sets in N and showed that these sets are partition regular and contain all linear partition regular configurations in N.  ...  Then for any finite partition of a Λ t -system, one of the cells in the partition is still a Λ t -system. Theorem 3.20 is proved in Section 6.  ...
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