Related papers: A Downward Collapse within the Polynomial Hierarch…
This paper presents the following results on sets that are complete for NP. 1. If there is a problem in NP that requires exponential time at almost all lengths, then every many-one NP-complete set is complete under length-increasing…
A poset can be regarded as a category in which there is at most one morphism between objects, and such that at most one of Hom(c,c') and Hom(c',c) is nonempty for distinct objects c,c'. If we keep in place the latter axiom but allow for…
We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author's pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples…
A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…
The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axis-aligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
The \emph{strong collapse} of a simplicial complex, proposed by Barmak and Minian (\emph{Disc. Comp. Geom. 2012}), is a combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention from computational…
We discuss a general method by which a higher order difference equation on a group is transformed into an equivalent triangular system of two difference equations of lower orders. This breakdown into lower order equations is based on the…
The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height…
Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…
We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is…
Reduction trees are a way of encoding a substitution procedure dictated by the relations of an algebra. We use reduction trees in the subdivision algebra to construct canonical triangulations of flow polytopes which are shellable. We…
In all well-studied $\mathsf{TFNP}$ subclasses (e.g. $\mathsf{PPA}, \mathsf{PPP}$ etc.), the canonical complete problem takes as input a polynomial-size circuit $C: \{ 0, 1\}^n \rightarrow \{ 0, 1\}^m$ whose input-output behavior implicitly…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
Monsky's celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different…
A classical result from topology called Uryshon's lemma asserts the existence of a continuous separator of two disjoint closed sets in a sufficiently regular topological space. In this work we make a search for this separator constructive…
Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…
Homothetic scalar field collapse is considered in this article. By making a suitable choice of variables the equations are reduced to an autonomous system. Then using a combination of numerical and analytic techniques it is shown that there…