Orbits Of Polynomials Research Articles

Equivariant algebraic and semi-algebraic geometry of infinite affine space

We study Sym(∞)-orbit closures of non-necessarily closed points in the Zariski spectrum of the infinite polynomial ring C[xij:i∈N,j∈[n]]. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by Sym(∞) orbits of polynomials in R[xij:i∈N,j∈[n]]. For n=1 we prove a quantifier elimination type result which fails for n>1.

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

The orbit of an n -variate polynomial \(f({\mathbf {x}})\) over a field \(\mathbb {F}\) is the set \(\text {orb}(f) := \lbrace f(A{\mathbf {x}}+{\mathbf {b}}) : A \in \mathrm{GL}(n,\mathbb {F}) \text{ and } {\mathbf {b}}\in \mathbb {F}^n\rbrace\) . The orbit of a polynomial f is a geometrically interesting subset of the set of affine projections of f . Affine projections of polynomials computable by seemingly weak circuit classes can be quite powerful. For example, the polynomial \(\mathsf {IMM}_{3,d}\) —the \((1,1)\) th entry of a product of d generic \(3 \times 3\) matrices—is computable by a constant-width read-once oblivious algebraic branching program (ROABP), yet every polynomial computable by a size- s general arithmetic formula is an affine projection of \(\mathsf {IMM}_{3,\text {poly}(s)}\) as shown by Ben-or and Cleve [ 12 ]. To our knowledge, no efficient hitting set construction was known for \(\text {orb}(\mathsf {IMM}_{3, d})\) before this work. In this article, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: Low-individual-degree polynomials computable by commutative ROABPs . This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials and the orbits of multilinear sparse polynomials . Multilinear polynomials computable by constant-width ROABPs . This implies a quasi-polynomial time hitting set for the orbits of the family \(\lbrace \mathsf {IMM}_{3,d}\rbrace _{d \in \mathbb {N}}\) . Polynomials computable by constant-depth, constant-occur formulas . This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in , and also polynomial-time hitting sets for the orbits of the power symmetric polynomials and the sum-product polynomials . Polynomials computable by occur-once formulas . We say a polynomial has low individual degree if the degree of every variable in the polynomial is at most \(\text {poly}(\log n)\) , where n is the number of variables. The first two results are obtained by building upon and strengthening the rank concentration by translation technique of Agrawal, Saha, and Saxena [ 6 ]; the second result additionally uses the merge-and-reduce idea from Forbes and Shpilka [ 30 ] and Forbes, Shpilka, and Saptharishi [ 27 ]. The proof of the third result adapts the algebraic independence-based technique of Agrawal, Saha, Saptharishi, and Saxena [ 5 ] and Beecken, Mittmann, and Saxena [ 11 ] to reduce to the case of constructing hitting sets for the orbits of sparse polynomials. A similar reduction using the Shpilka-Volkovich (SV) generator-based argument in Shpilka and Volkovich [ 90 ] yields the fourth result. The SV generator plays an important role in all four results.

Arithmetic progressions in polynomial orbits

Let f be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit [Formula: see text], where t is an integer, using arithmetic progressions each of which contains t. Fixing an integer [Formula: see text], we prove that it is impossible to cover [Formula: see text] using k such arithmetic progressions unless [Formula: see text] is contained in one of these progressions. In fact, we show that the relative density of terms covered by k such arithmetic progressions in [Formula: see text] is uniformly bounded from above by a bound that depends solely on k. In addition, the latter relative density can be made as close as desired to 1 by an appropriate choice of k arithmetic progressions containing t if k is allowed to be large enough.

Polynomial orbits in totally minimal systems

Inspired by the recent work of Glasner, Huang, Shao, Weiss and Ye [14], we prove that the maximal ∞-step pro-nilfactor X∞ of a minimal system (X,T) is the topological characteristic factor along polynomials in a certain sense. Namely, we show that by an almost one to one modification of π:X→X∞, the induced open extension π⁎:X⁎→X∞⁎ has the following property: for any d∈N, any open subsets V0,V1,…,Vd of X⁎ with ⋂i=0dπ⁎(Vi)≠∅ and any distinct non-constant integer polynomials pi with pi(0)=0 for i=1,…,d, there exists some n∈Z such that V0∩T−p1(n)V1∩…∩T−pd(n)Vd≠∅.As an application, the following result is obtained: for a totally minimal system (X,T) and integer polynomials p1,…,pd, if every non-trivial integer combination of p1,…,pd is not constant, then there is a dense Gδ subset Ω of X such that the set{(Tp1(n)x,…,Tpd(n)x):n∈Z} is dense in Xd for every x∈Ω.

Discrete Analogues in Harmonic Analysis: Directional Maximal Functions in ℤ2

Abstract Let $V = \{ v_1,\dots ,v_N\}\subset \mathbb{Z}^2$ be a collection of $N$ vectors that lie near a discrete sphere. We consider discrete directional maximal functions on $\mathbb{Z}^2$ where the set of directions lies in $V$, given by $$\begin{align*}& \sup_{v \in V, k \geq C \log N} \left| \sum_{n \in \mathbb{Z}} f(x-nv ) \phi_k(n) \right|, \ f:\mathbb{Z}^2 \to \mathbb{C},\end{align*}$$where $\phi _k(t):= 2^{-k} \phi (2^{-k} t)$ for some bump function $\phi$. Interestingly, the study of these operators leads one to consider an “arithmetic version” of a Kakeya-type problem in the plane, which we approach using a combination of geometric and number-theoretic methods. Motivated by the Furstenberg problem from geometric measure theory, we also consider a discrete directional maximal operator along polynomial orbits, $$\begin{align*}& \sup_{v \in V} \left| \sum_{n \in \mathbb{Z}} f(x-v\cdot P(n) ) \cdot \phi(n) \right|, \ P \in \mathbb{Z}[-].\end{align*}$$

On semigroup orbits of polynomials in multiplicative subgroups

On semigroup orbits of polynomials in multiplicative subgroups

ON SEMIGROUP ORBITS OF POLYNOMIALS AND MULTIPLICATIVE ORDERS

We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski [‘Multiplicative orders in orbits of polynomials over finite fields’, Glasg. Math. J.60(2) (2018), 487–493].

Unlikely intersections over finite fields: Polynomial orbits in small subgroups

We estimate the frequency of polynomial iterations which fall in a given multiplicative subgroup of a finite field of \begin{document}$ p $\end{document} elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first \begin{document}$ N $\end{document} elements in an orbit. We derive these from more general results about sequences of compositions on a fixed set of polynomials.

MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS

AbstractWe show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

Linear relations between polynomial orbits

We study the orbits of a polynomial f∈C[X], namely, the sets {α,f(α),f(f(α)),…} with α∈C. We prove that if two nonlinear complex polynomials f,g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in Cd with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell–Lang conjecture.

The quantitative behaviour of polynomial orbits on nilmanifolds

A theorem of Leibman asserts that a polynomial orbit (g(n)G) n?Z on a nilmanifold G/G is always equidistributed in a union of closed sub-nilmanifolds of G/G . In this paper we give a quantitative version of Leibman�s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)G) n?[N] in a nilmanifold. More specifically we show that there is a factorisation g=eg ' ? , where e(n) is �smooth,� (?(n)G) n?Z is periodic and �rational,� and (g ' (n)G) n?P is uniformly distributed (up to a specified error d ) inside some subnilmanifold G ' /G ' of G/G for all sufficiently dense arithmetic progressions P?[N] . Our bounds are uniform in N and are polynomial in the error tolerance d . In a companion paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from an earlier paper of ours.

Primitive Prime Divisors in Zero Orbits of Polynomials

Abstract.Let

Intersections of polynomial orbits, and a dynamical Mordell–Lang conjecture

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture.

Finite polynomial orbits in quadratic rings

All finite orbits of polynomial mappings in one variable in rings of integers of quadratic number fields are determined. The largest such orbit has seven elements and lies in the third cyclotomic field.

Polynomial orbits in finite commutative rings

Let R be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.

Scarcity of finite polynomial orbits

Scarcity of finite polynomial orbits

Finite polynomial orbits in finitely generated domains

LetR be a commutative domain of zero characteristic and letf(X) be a polynomial with coefficients inR. It is shown that all finite orbits inR. under the mapping induced byR have their cardinalities bounded by a constant depending onR but not onf. In the case whenR is the ring of all integers in an algebraic number field this constant is effectively determined.