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We extend Quine's bound on the number of self-intersection of curves with polynomial parameterization to the case of Laurent polynomials. As an application, we show that circle embeddings are dense among all maps from a circle to a plane with respect to an integral norm., Comment: 10 pages

We consider paths of steepest descent, in the complex plane, for the norm of a non-constant one variable polynomial $f$. We show that such paths, starting from a zero of the logarithmic derivative of $f$ and ending in a root of $f$, draw a tree in the complex plane, and we give an upper bound estimate on their lengths. In some cases, we obtain a finer estimate that depends only on the set of roots of $f$, not on their multiplicity, and we wonder if this can be done in general. We also extend this question to finite Blaschke products for the unit disk., Comment: Several typos corrected, addition of a section on Blaschke products, 7 pages, 2 figures

We study the relationship between polynomial approximations in the Bergman space of certain simply connected domains in the complex plane and composition operators on the Dirichlet space of the unit disk. In particular, we characterize when a composition operator on the Dirichlet space has dense range, which settles a problem posed by Joseph Cima in 1976.

Let p n : C → C p_n:\mathbb {C} \rightarrow \mathbb {C} be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution 2 π r u ( r ) d r 2\pi r u(r) dr in the complex plane. A natural question is how the distribution of roots evolves under repeated (say n / 2 − n/2- times) differentiation of the polynomial. We conjecture a mean-field expansion for the evolution of ψ ( s ) = u ( s ) s \psi (s) = u(s) s : ∂ ψ ∂ t = ∂ ∂ x ( ( 1 x ∫ 0 x ψ ( s ) d s ) − 1 ψ ( x ) ) . \begin{equation*} \frac {\partial \psi }{\partial t} = \frac {\partial }{\partial x} \left ( \left ( \frac {1}{x} \int _{0}^{x} \psi (s) ds \right )^{-1} \psi (x) \right ). \end{equation*} The evolution of ψ ( s ) ≡ 1 \psi (s) \equiv 1 corresponds to the evolution of random Taylor polynomials p n ( z ) = ∑ k = 0 n γ k z k k ! where γ k ∼ N C ( 0 , 1 ) . \begin{equation*} p_n(z) = \sum _{k=0}^{n}{ \gamma _k \frac {z^k}{k!}} \quad \text {where} \quad \gamma _k \sim \mathcal {N}_{\mathbb {C}}(0,1). \end{equation*} We discuss some numerical examples suggesting that this particular solution may be stable. We prove that the solution is linearly stable. The linear stability analysis reduces to the classical Hardy integral inequality. Many open problems are discussed.

We realize any submultiplicative increasing function which is equivalent to a polynomial proportion of itself as the growth function of a finitely generated simple Lie algebra. As an application, we resolve two open problems posed by Petrogradsky on Lie algebras with intermediate growth., Accepted for publication in Proc. AMS

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $\theta$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb{C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb{C}^\ell \to \mathbb{C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture., Comment: 13 pages

Gasull and Sabatini in [Ann. Mat. Pura Appl. 198 (2019), pp. 1985–2006] studied limit cycles of a Liénard system which has a fixed invariant curve, i.e., a Wilson polynomial Liénard system. The Liénard system can be changed into x ˙ = y − ( x 2 − 1 ) ( x 3 − b x ) , y ˙ = − x ( 1 + y ( x 3 − b x ) ) \dot x=y-(x^2-1)(x^3-bx), ~ \dot y=-x(1+y(x^3-bx)) . For b ≤ 0.7 b\leq 0.7 and b ≥ 0.76 b\geq 0.76 , limit cycles of the system are studied completely. But for 0.7 > b > 0.76 0.7>b>0.76 , the exact number of limit cycles is still unknown, and Gasull and Sabatini conjectured that the exact number of limit cycles is two (including multiplicities). In this paper, we give a positive answer to this conjecture and study all bifurcations of the system. Finally, we show the expanding of the moving limit cycle as b > 0 b>0 increases and give all global phase portraits on the Poincaré disk of the system completely.

The simplest version of Bertini’s irreducibility theorem states that the generic fiber of a noncomposite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if f f is a noncommutative polynomial such that f − λ f-\lambda factors for infinitely many scalars λ \lambda , then there exist a noncommutative polynomial h h and a nonconstant univariate polynomial p p such that f = p ∘ h f=p\circ h . Two applications of free Bertini’s theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of f f is the set of all matrix tuples X X where f ( X ) f(X) attains some given eigenvalue. It is shown that eigenlevel sets of f f and g g coincide if and only if f a = a g fa=ag for some nonzero noncommutative polynomial a a . The second application pertains to quasiconvexity and describes polynomials f f such that the connected component of \{X \text { tuple of symmetric n×n matrices}\colon \lambda I\succ f(X) \} about the origin is convex for all natural n n and λ > 0 \lambda >0 . It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial.

Robert L. Bryant (see [Astérisque 154-155 (1987), pp. 321–347, Proposition 4]) proved that an isolated singularity of a conformal metric of positive constant curvature with finite area on a Riemann surface is a conical one. Using complex analysis, we find all local models for an isolated singularity of a flat metric whose area satisfies some polynomial growth condition near the singularity. In particular, we show that an isolated singularity of a flat metric with finite area must be conic.

It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any compact subset $K$ of a Euclidean space, there exists a set $X$, in some ${\mathbb C}^N$, that is homeomorphic to $K$ and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in ${\mathbb C}^3$ constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown also for $m$-dimensional manifolds ($m\geq 2$) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold., Comment: Some results have been strengthened by decreasing the dimension of the ambient space

We give conditions under which the number of solutions of a system of polynomial equations over a finite field F_q of characteristic p is divisible by p. Our setup involves the substitution t_i |-> f_i(t_i) for auxiliary polynomials f_1,...,f_n in F_q[t]. We recover as special cases results of Chevalley-Warning and Morlaye-Joly. Then we investigate higher p-adic divisibilities, proving a result that recovers the Ax-Katz Theorem. We also consider p-weight degrees, recovering work of Moreno-Moreno, Moreno-Castro and Castro-Castro-Velez., 15 pages

Recently, it was observed that the roots of the Askey-Wilson polynomial are retrieved at the unique global minimum of an associated strictly convex Morse function [J. F. van Diejen and E. Emsiz, Lett. Math. Phys. 109 (2019), pp. 89–112]. The purpose of the present note is to infer that the corresponding gradient flow converges to the roots in question at an exponential rate.

In this note, a general result for determining the rational hulls of fibered sets in C 2 \mathbb {C}^2 is established. We use this to compute the rational hull of Rudin’s Klein bottle, the first explicit example of a totally real nonorientable surface in C 2 \mathbb {C}^2 . In contrast to its polynomial hull, which was shown to contain an open set by the first author in 2012, its rational hull is shown to be 2 2 -dimensional. Using the same method, we also compute the rational hulls of some other surfaces in C 2 \mathbb {C}^2 .

An $n\times n$ sign pattern $S$, which is a matrix with entries $0,+,-$, is called spectrally arbitrary if any monic real polynomial of degree $n$ can be realized as a characteristic polynomial of a matrix obtained by replacing the non-zero elements of $S$ by numbers of the corresponding signs. A sign pattern $S$ is said to be a superpattern of those matrices that can be obtained from $S$ by replacing some of the non-zero entries by zeros. We develop a new technique that allows us to prove spectral arbitrariness of sign patterns for which the previously known "Nilpotent Jacobian" method does not work. Our approach leads us to solutions of numerous open problems known in the literature. In particular, we provide an example of a sign pattern $S$ and its superpattern $S'$ such that $S$ is spectrally arbitrary but $S'$ is not, disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den Driessche., 5 pages

We will show that ( 1 − q ) ( 1 − q 2 ) … ( 1 − q m ) (1-q)(1-q^2)\dots (1-q^m) is a polynomial in q q with coefficients from { − 1 , 0 , 1 } \{-1,0,1\} iff m = 1 , 2 , 3 , m=1,\ 2,\ 3, or 5 and explore some interesting consequences of this result. We find explicit formulas for the q q -series coefficients of ( 1 − q 2 ) ( 1 − q 3 ) ( 1 − q 4 ) ( 1 − q 5 ) … (1-q^2)(1-q^3)(1-q^4)(1-q^5)\dots and ( 1 − q 3 ) ( 1 − q 4 ) ( 1 − q 5 ) ( 1 − q 6 ) … (1-q^3)(1-q^4)(1-q^5)(1-q^6)\dots . In doing so, we extend certain observations made by Sudler in 1964. We also discuss the classification of the products ( 1 − q ) ( 1 − q 2 ) … ( 1 − q m ) (1-q)(1-q^2)\dots (1-q^m) and some related series with respect to their absolute largest coefficients.

Let $\Gamma$ be a graph and $P$ be a reversible random walk on $\Gamma$. From the $L^2$ analyticity of the Markov operator $P$, we deduce that an iterate of odd exponent of $P$ is `lazy', that is there exists an integer $k$ such that the transition probability (for the random walk $P^{2k+1}$) from a vertex $x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $\Gamma$ but only a polynomial control of the volume., Comment: 12 pages

We improve the Cauchy radius of both scalar and matrix polynomials, which is an upper bound on the moduli of the zeros and eigenvalues, respectively, by using appropriate polynomial multipliers., Comment: 12 pages

The goal of the present note is to construct a class of examples for connected compact sets K subset of C-n whose polynomial hull (K) over cap cannot be covered by analytic discs with boundaries co ...

We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not grow exponentially fast with the dimension. Many classical problems are known to be intractable. However, sometimes tractability can be shown. Often such proofs are based on randomly selected integration nodes. Of course, in applications, true random numbers are not available and hence one mimics them with pseudorandom number generators. This motivates us to propose the use of pseudorandom vectors as underlying integration nodes in order to achieve tractability. In particular, we consider digital inverse vectors and present two examples of problems, the weighted star discrepancy and integration of Hölder continuous, absolute convergent Fourier and cosine series, where the proposed method is successful.

We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.

We describe a method for bounding the cyclicity of the class of monodromic singularities of polyn omial planar families of vector fields X λ with an analytic Poincar e first return map having a polynomial Bautin ideal B in the ring of polynomials in the parameters λ of the family. This class includes the nondegenerate centers, generic nilpotent centers and also some degenerate centers. This method can work even in the case in which B is not radical by studying the stabilization of the integral closures of an ascending chain of polynomial ideals that stabilizes at B. The approach is based on computational algebra methods for determining a minimal basis of the integral closurē B of B. As far as we know, the obtained cyclicity bound is the minimum found in the literature. The first author was partially supported by a MINECO grant number MTM2014-53703-P and by a CIRIT grant number 2014 SGR 1204.

Given a multi-variable polynomial, there is an associated divided symmetrization (in particular turning it into a symmetric function). Postinkov has found the volume of a permutohedron as a divided symmetrization (DS) of the power of a certain linear form. The main task in this paper is to exhibit and prove closed form DS-formulas for a variety of polynomials. We hope the results to be valuable and available to the research practitioner in these areas. Also, the methods of proof utilized here are simple and amenable to many more analogous computations. We conclude the paper with a list of such formulas., Comment: 13 pages, no figures

We give a variant of the Bohenblust-Hille inequality which, for certain families of polynomials, leads to constants with polynomial growth in the degree., The third author was supported by MICINN MTM2011-22417 and UPV-SP20120700.

We study the relation between left and right adjoint functors to the precomposition functor. As a consequence, we obtain various dualities for the Ext–groups in the category of strict polynomial functors.

McNamara and Sagan conjectured that if a0, a1, a2, . . . is a Polya frequency (PF) sequence, then so is (formula presented), . . .. We prove this conjecture for a natural class of PF-sequences whic ...

We analyze the weighted star discrepancy of so-called $p$-sets which go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have largely been ignored since a number of other constructions have been discovered which achieve a better convergence rate. However, it has recently been discovered that the $p$-sets perform well in terms of the dependence on the dimension. We prove bounds on the weighted star discrepancy of the $p$-sets which hold for any choice of weights. For product weights we give conditions under which the discrepancy bounds are independent of the dimension $s$. This implies strong polynomial tractability for the weighted star discrepancy. We also show that a very weak condition on the product weights suffices to achieve polynomial tractability.

In this paper we prove a $q$-analogue of Koshy's formula in terms of the Narayana polynomial due to Lassalle and a $q$-analogue of Koshy's formula in terms of $q$-hypergeometric series due to Andrews by applying the inclusion-exclusion principle on Dyck paths and on partitions. We generalize these two $q$-analogues of Koshy's formula for $q$-Catalan numbers to that for $q$-Ballot numbers. This work also answers an open question by Lassalle and two questions raised by Andrews in 2010. We conjecture that if $n$ is odd, then for $m\ge n\ge 1$, the polynomial $(1+q^n){m\brack n-1}_q$ is unimodal. If $n$ is even, for any even $j\ne 0$ and $m\ge n\ge 1$, the polynomial $(1+q^n)[j]_q{m\brack n-1}_q$ is unimodal. This implies the answer to the second problem posed by Andrews.

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

We give a general reduction of lengths-of-proofs lower bounds for constant depth Frege systems in DeMorgan language augmented by a connective counting modulo a prime p (the so called AC[p] Frege systems) to computational complexity lower bounds for search tasks involving search trees branching upon values of linear maps on the vector space of low degree polynomials over Fp. In 1988 Ajtai [2] proved that the unsatisfiable set (¬PHPn) of propositional formulas ∨ j∈[n] pij and ¬pi1j ∨ ¬pi2j and ¬pij1 ∨ ¬pij2 for all i ∈ [n + 1] = {1, . . . , n + 1} , all i1 6= i2 ∈ [n + 1], j ∈ [n], and all i ∈ [n + 1], j1 6= j2 ∈ [n] respectively, expressing the failure of the pigeonhole principle (PHP), has for no d ≥ 1 a polynomial size refutation in a Frege proof system operating only with DeMorgan formulas of depth at most d. Subsequently Krajicek [18] established an exponential lower bound for these so called AC Frege proof systems (for different formulas) and Krajicek, Pudlak and Woods [23] and Pitassi, Beame and Impagliazzo [26] improved independently (and announced jointly in [7]) Ajtai’s bound for PHP to exponential. All these papers employ some adaptation of the random restriction method that has been so successfully applied earlier in circuit complexity (cf. [1, 14, 31, 15]). Razborov [28] invented already in 1987 an elegant method, simplified and generalized by Smolensky [30], to prove lower bounds even for AC[p] circuits, p a prime. Thus immediately after the lower bounds for AC Frege systems were proved researchers attempted to adapt the Razborov-Smolensky method to proof complexity and to prove lower bounds also for AC[p] Frege systems. This turned out to be rather elusive and no lower bounds for the systems were proved, although some related results were obtained. Ajtai [3, 4, 5], Beame et.al.[6] and Buss et.al.[9] proved lower bounds for AC Frege systems in DeMorgan language augmented by the so called modular counting principles as extra axioms (via degree lower bounds for the Nullstellensatz proof system in [6, 9])

Motivated by a conjecture of Savage and Visontai about the equidistribution of the descent statistic on signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$ and the ascent statistic on $(1,4,3,8,\ldots,2n-1,4n)$-inversion sequences, we investigate the descent polynomial of the signed permutations of a general multiset. We obtain a factorial generating function formula for a $q$-analog of these descent polynomials and apply it to show that they have only real roots. Two different proofs of the conjecture of Savage and Visontai are provided.

Several questions in complex analysis lead naturally to the study of bihomogeneous polynomials r(z, z) on Cn×Cn for which r(z, z) ‖z‖ = ‖h(z)‖ for some natural number d and a holomorphic polynomial mapping h = (h1, . . . , hK) from Cn to CK . When r has this property for some d, one seeks relationships between d, K, and the signature and rank of the coefficient matrix of r. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in C[z1, . . . , zn] and apply a well-known result of Macaulay to estimate some natural quantities.

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted $L^2$ condition on the norm of the second fundamental form. Our approach adopt the viewpoint of weighted manifolds and permits also to recover and to extend some others recent classification and gap results for self-shrinkers., 9 pages. To appear on Proc. Amer. Math. Soc

We discuss several aspects of Łojasiewicz inequalities, namely local and global versions, and the relations between the gradient inequality and regular separation of real algebraic sets. We give effective estimates for Łojasiewicz’s exponents in the real and complex setting.

Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result of this paper is a proof of Sendov conjecture when the polynomial $P$ has a degree higher than a fixed integer $N$. We will give estimates of its integer $N$ in terms of $|a|$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P'$) of a polynomial which would contradict Sendov conjecture., Comment: 14 pages, 5 figures

Consider a representation $\rho\colon L \to \mathfrak{gl}(V)$ where $L$ is a Lie algebra and $V$ is a finite dimensional vector space. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities of $\rho$., Comment: 12 pages. Lemma 11 and Lemma 12 were clarified. Also some other minor misprints has been corrected. Examples 1-5 were added

For a closed n-braid L with a full positive twist and with k negative crossings, 0\leq k \leq n, we determine the first n-k+1 terms of the Jones polynomial V_L(t). We show that V_L(t) satisfies a braid index constraint, which is a gap of length at least n-k between the first two nonzero coefficients of (1-t^2)V_L(t). For a closed positive n-braid with a full positive twist, we extend our results to the colored Jones polynomials. For N>n-1, we determine the first n(N-1)+1 terms of the normalized N-th colored Jones polynomial., 13 pages.

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space., References are updated

Central sets in N were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set con- tains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of ho- mogeneous linear equations. We introduce here the notions of strongly central and very strongly central, which is as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than syndetic, which in the case of N means that gaps are bounded. Given x 2 R, let w(x) = x b x + 1 c. Kronecker's Theorem says that if 1, 1, 2,..., v are linearly independent over Q and U is a nonempty open subset of ( 1 , 1 ) v , then {x 2 N : (w( 1x),...,w( vx)) 2 U} is nonempty and Weyl showed that this set has positive density. We show here that if 0 is in the closure of U, then this set is strongly central. More generally, let P1,P2,...,Pv be real polynomials with zero constant term. We show that {x 2 N : (w(P1(x)),...,w(Pv(x))) 2 U} is nonempty for every open U with 0 2 c'U if and only if it is very strongly central for every such U and we show that these conclusions hold if and only if any nontrivial rational linear combination of P1,P2,...,Pv has at least one irrational coecient.

We find bounds on the Hilbert space compression exponent of the limit of a directed metric system of groups. We also give estimates on the Hilbert space compression exponent of a group extension of a group H H by a word-hyperbolic group or a group of polynomial growth.

We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero., Comment: 12 pages

We study the factorization into irreducibles of iterates of a quadratic polynomial f f over a finite field. We call f f settled when the factorization of its n n th iterate for large n n is dominated by “stable” polynomials, namely those that are irreducible under post-composition by any iterate of f f . We prove that stable polynomials may be detected by their action on the critical orbit of f f and that the critical orbit also gives information about the splitting of non-stable polynomials under post-composition by iterates of f f . We then define a Markov process based on the critical orbit of f f and conjecture that its limiting distribution describes the full factorization of large iterates of f f . This conjecture implies that almost all quadratic f f defined over a finite field are settled. We give several types of evidence for our conjecture.

We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type f ( I ) ⋅ H f(I)\cdot H are equidistributed (as p p tends to infinity), where f f is a polynomial, I I is an interval of F p \mathbb {F}_p and H H is an approximate subgroup of F p ∗ \mathbb {F}_p^* with size larger than polylogarithmic in p p .

Let K be a global field and let O K,S be the ring of S-integers of K for some finite set S of primes of K. We prove that whatever the infinite subset E C O K,S and the polynomial f(X) ∈ K[X], the subsets E and f(E) have the same number of residual classes modulo m for almost all maximal ideals m of O K,S if and only if deg(f) = 1 when the characteristic of K is 0 and f(X) = g(X pk ) for some integer k and some polynomial g with deg(g) = 1 when the characteristic of K is p > 0.

In the present paper, we will derive the Liouville theorem and the finite dimension theorem for polynomial growth harmonic functions defined on Alexandrov spaces with nonnegative Ricci curvature in the sense of Kuwae-Shioya and Sturm-Lott-Villani.

In this paper, we prove the presence of limit cycles of given multi- plicity, together with a complete unfolding, in families of (singularly perturbed) polynomial Li enard equations. The obtained limit cycles are relaxation oscil- lations. Both classical Li enard equations and generalized Li enard equations are treated.