A STABILITY VERSION OF THE GAUSS–LUCAS THEOREM AND APPLICATIONS
Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$ , are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk, $$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$ then $p^{\prime }$ has $n-1$ roots inside the unit disk and $m$ roots at distance at least $(dn-m)/(n+m)>1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for $n$ sufficiently large, each of the $m$ roots has a critical point at distance ${\sim}n^{-1}$ .
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10133832
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Journal of the Australian Mathematical Society
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1 to 8
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1446-7887
Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in ${\mathbb{R}}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$ such that each manifold $Z_d$ is diffeomorphic to a component of the zero set on $\Gamma$ of some polynomial of degree $d$. (This is in sharp contrast with the case when $\Gamma$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $p$ on $\Gamma$ is bounded by a polynomial in $\deg (p)$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$ containing a subset $D$ homeomorphic to a disk, and a family of polynomials $\{p_m\}_{m\in \mathbb{N}}$ of degree $\deg (p_m)=d_m$ such that $(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$ i.e. the zero set of $p_m$ in $D$ is isotopic to $Z_{d_m}$ in ${\mathbb{R}}^{n-1}$. This says that, up to extracting subsequences, the intersection of $\Gamma$ with a hypersurface of degree $d$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $0 \leq k \leq n-2$ and every sequence of natural numbers $a=\{a_d\}_{d\in \mathbb{N}}$ there is a regular, compact semianalyticmore »
Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$${a}_{j,i}\in {F}_{p}$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$${a}_{j,1}+\cdots +{a}_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m×m$minor of the$$m\times k$$$m×k$matrix$$(a_{j,i})_{j,i}$$${\left({a}_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq {F}_{p}^{n}$of size$$|A|> C\cdot \Gamma ^n$$$|A|>C·{\Gamma }^{n}$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$${x}_{1},\cdots ,{x}_{k}\in A$are all distinct. Here,Cand$$\Gamma$$$\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma . The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$in the solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. 3. The classic graphical Cheeger inequalities state that if M is an n\times n \emph{symmetric} doubly stochastic matrix, then $\frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))}$ where \phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right) is the edge expansion of M, and \lambda_{2}(M) is the second largest eigenvalue of M. We study the relationship between \phi(A) and the spectral gap 1-\re\lambda_{2}(A) for \emph{any} doubly stochastic matrix A (not necessarily symmetric), where \lambda_{2}(A) is a nontrivial eigenvalue of A with maximum real part. Fiedler showed that the upper bound on \phi(A) is unaffected, i.e., \phi(A)\leq\sqrt{2\cdot(1-\re\lambda_{2}(A))}. With regards to the lower bound on \phi(A), there are known constructions with $\phi(A)\in\Theta\left(\frac{1-\re\lambda_{2}(A)}{\log n}\right),$ indicating that at least a mild dependence on n is necessary to lower bound \phi(A). In our first result, we provide an \emph{exponentially} better construction of n\times n doubly stochastic matrices A_{n}, for which $\phi(A_{n})\leq\frac{1-\re\lambda_{2}(A_{n})}{\sqrt{n}}.$ In fact, \emph{all} nontrivial eigenvalues of our matrices are 0, even though the matrices are highly \emph{nonexpanding}. We further show that this bound is in the correct range (up to the exponent of n), by showing that for any doubly stochastic matrix A, $\phi(A)\geq\frac{1-\re\lambda_{2}(A)}{35\cdot n}.$ As a consequence, unlike the symmetric case, there is a (necessary) loss of amore » 4. We examine correlations of the Möbius function over \mathbb{F}_{q}[t] with linear or quadratic phases, that is, averages of the form 1$$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$if$Q$is linear and$O(q^{-n^{c}})$for some absolute constant$c>0$if$Q$is quadratic. The latter bound may be reduced to$O(q^{-c^{\prime }n})$for some$c^{\prime }>0$when$Q(f)$is a linear form in the coefficients of$f^{2}$, that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem. 5. In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence${\mathcal{D}}$, we obtain a sharp criterion such that for almost every$\unicode[STIX]{x1D6FC}$the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions$(n,p)\in \mathbb{N}\times \mathbb{Z}$for a certain one-parameter family of$\unicode[STIX]{x1D713}$. Also, under a minor condition on pseudo-absolute-value sequences${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$, we obtain a sharp criterion on a general sequence$\unicode[STIX]{x1D713}(n)$such that for almost every$\unicode[STIX]{x1D6FC}$the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions$(n,p)\in \mathbb{N}\times \mathbb{Z}\$ .