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Tamar Ziegler

Publications and source records attributed to Tamar Ziegler.

At least 19 recordsLinked to original sources
arXiv2025

Polynomial Bounds for Birch's Theorem

Let $K$ be a number field and $f_1,\ldots,f_s\in K[x_1,\ldots,x_n]$ forms of odd degrees. In 1957, Birch proved that if $n$ is sufficiently large then the forms always have a nontrivial zero in $K^n$. Apart from some small degrees, the number of variables required was so large that it has been described as "not even astronomical". We prove that, for any fixed degree, $n$ may be taken polynomial in $s$. We deduce this from a stronger result -- the Zariski closure of the set of rational zeros has codimension bounded by a polynomial in $s$. When $K$ is totally imaginary, our results hold for forms of any (possibly even) degrees.

math.NT↗
arXiv2025

Strength and partition rank under limits and field extensions

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop under field extensions, and both can jump in a limit. We show that, for fixed degree and under mild conditions on the characteristic of the ground field, the strength is at most a polynomial in the border strength. We also establish an analogous result for partition rank. Our results control both the jump under limits and the drop under field extensions.

math.AG↗
arXiv2024

Relative Rank and Regularization

We introduce a new concept of rank - relative rank associated to a filtered collection of polynomials. When the filtration is trivial our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case we obtain a new proof of the results in arXiv:1902.09830). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry and algebra. For example, we prove that any collection of polynomials $\mathcal{P}=(P_i)_{i=1}^c$ of degrees $\le d$ in a polynomial ring over an algebraically closed field of characteristic $>d$ is contained in an ideal $\mathcal{I}(\mathcal{Q})$, generated by a collection $\mathcal{Q}$ of polynomials of degrees $\le d$ which form a regular sequence, and $\mathcal{Q}$ is of size $\le A c^{A}$, where $A=A(d)$ is independent of the number of variables.

math.AC↗
arXiv2024

A Dense Model Theorem for the Boolean Slice

The (low soundness) linearity testing problem for the middle slice of the Boolean cube is as follows. Let $\varepsilon>0$ and $f$ be a function on the middle slice on the Boolean cube, such that when choosing a uniformly random quadruple $(x,y,z ,x\oplus y\oplus z)$ of vectors of $2n$ bits with exactly $n$ ones, the probability that $f(x\oplus y \oplus z) = f(x) \oplus f(y) \oplus f(z)$ is at least $1/2+\varepsilon$. The linearity testing problem, posed by David, Dinur, Goldenberg, Kindler and Shinkar, asks whether there must be an actual linear function that agrees with $f$ on $1/2+\varepsilon'$ fraction of the inputs, where $\varepsilon' = \varepsilon'(\varepsilon)>0$. We solve this problem, showing that $f$ must indeed be correlated with a linear function. To do so, we prove a dense model theorem for the middle slice of the Boolean hypercube for Gowers uniformity norms. Specifically, we show that for every $k\in\mathbb{N}$, the normalized indicator function of the middle slice of the Boolean hypercube $\{0,1\}^{2n}$ is close in Gowers norm to the normalized indicator function of the union of all slices with weight $t = n\pmod{2^{k-1}}$. Using our techniques we also give a more general `low degree test' and a biased rank theorem for the slice.

math.CO↗
arXiv2024

On rank in algebraic closure

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with $R_i, S_i \in {\mathbf k}[x_1, \ldots, x_s]$ forms of degree $ 4$. This result has immediate consequences for counting integer points (when $ {\mathbf k} $ is a number field) or prime points (when $ {\mathbf k} = \mathbb Q $) of the variety $ \{Q=0\} $ assuming $ rk_{\mathbf k} (Q) $ is large.

math.NT↗
arXiv2024

Infinite partial sumsets in the primes

We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.

math.NT↗
arXiv2022

Higher uniformity of bounded multiplicative functions in short intervals on average

Let $λ$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} λ(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for all fixed $k$ and $X^θ \leq H \leq X$ with $0 < θ< 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. We obtain this result as a special case of the corresponding statement for (non-pretentious) $1$-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases $e(-P(n))$ by degree $k$ nilsequences $\overline{F}(g(n) Γ)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$\int_{X}^{2X} \| λ\|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ in the same range of $H$. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of $λ$ over short polynomial progressions $(n+P_1(m),\ldots, n+P_k(m))$, which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.

math.NT↗
arXiv2020

Properties of high rank subvarieties of affine spaces

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties of high rank systems of polynomials. In the second, we use these properties to deduce results in Algebraic Geometry, such as an effective Stillman conjecture over algebraically closed fields, an analogue of Nullstellensatz for varieties over finite fields, and a strengthening of a recent result of [5]. We also show that for $k$-varieties $\mathbb X \subset \mathbb{A}^n$ of high rank any weakly polynomial function on a set $\mathbb{X}(k)\subset k^n$ extends to a polynomial.

math.AG↗
arXiv2020

On the codimension of the singular locus

Let $k$ be a field and $V$ an $k$-vector space. For a family $\bar P=\{ P_i\}_{1\leq i\leq c}, $ of polynomials on $V$, we denote by $\mathbb X _{\bar P}\subset V$ the subscheme defined by the ideal generated by $ \bar P$. We show the existence of $γ(c,d)$ such that the varieties $\mathbb X_{\bar P}$ are smooth outside of codimension $m$, if deg$(P_i)\leq d$ and rank (strength) $r_{nc}(\bar P)\geq γ(d,c) (1+m)^{γ(d,c)}$.

math.AG↗
arXiv2019

Extending weakly polynomial functions from high rank varieties

Let $k$ be a field, $V$ a $k$-vector space and $X$ be a subset of $V $. A function $f:X\to k$ is weakly polynomial of degree $\leq a$, if the restriction of $f$ on any affine subspace $L\subset X$ is a polynomial of degree $\leq a$. In this paper we consider the case when $X= \mathbb X (k)$ where $\mathbb X$ is a complete intersection of bounded codimension defined by a high rank polynomials of degrees $d, char(k)=0$ or $char (k)>d$ and either $k$ is algebraically closed, or $k=\mathbb F _q,q>ad$. We show that under these assumptions any $k$-valued weakly polynomial function of degree $ \leq a$ on $X$ is a restriction of a polynomial of degree $\leq a$ on $V$. Our proof is based on Theorem 1.11 on fibers of polynomial morphisms $P:\mathbb F _q^n\to \mathbb F _q^m$ of high rank. This result is of an independent interest. For example it immediately implies a strengthening of the result of [4].

math.AG↗
arXiv2018

Polynomial functions as splines

Let $V$ be a vector space over a finite field $k$. We give a condition on a subset $A \subset V$ that allows for a local criterion for checking when a function $f:A \to k$ is a restriction of a polynomial function of degree $<m$ on $V$. In particular, we show that high rank hypersurfaces of $V$ of degree $\ge m$ satisfy this condition. In addition we show that the criterion is robust (namely locally testable in the theoretical computer science jargon).

math.CO↗
arXiv2018

On ranks of polynomials

Let $V$ be a vector space over a field $k, P:V\to k, d\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\to k$ of degree $d$ such that $rank(\partial P/\partial t)\leq r$ for all $t\in V-0$. Our proof of this theorem is based on the application of results on Gowers norms for finite fields $k$. We don't know a direct proof in the case when $k=\mathbb C$.

math.AG↗
arXiv2017

Extending linear and quadratic functions from high rank varieties

Let $k$ be a field, $V$ be a $k$-vector space and $X\subset V$ an algebraic irreducible subvariety. We say that a function $f:X(k) \to k$ is weakly linear if its restriction to any two-dimensional linear subspace $W$ of $V$ contained in $X$ is linear and that it is weakly quadratic if its restriction to any three-dimensional linear subspace $W$ of $V$ contained in $X$ is quadratic. We say that $X$ is admissible if any weakly linear function on $X$ is a restriction of a linear function on $V$ and any weakly quadratic function on $X$ is a restriction of a quadratic function on $V$. The main result in the paper concerns the case when the field $k$ is a finite. We show that for any $d,L\geq 1$ there exists $r=r(d,L,k)\in \mathbb Z _+$ such that any complete intersection $X\in V$ in a vector space $V$ of codimension $L$, degree $d$ and rank $\geq r$ is admissible. Moreover we show the existence of a function $r(d,L)$ such that one can take $r(d,L,k)=r(d,L)$ for all finite fields $k$ of characteristic $>d$. The proof of the admissibility for finite fields $k$ is based on bounds on the number of $k$-points on ancillary varieties $E(X)$. These results allow us to bound the dimension of varieties $E(X)$. Using these results we were able to prove the admissibility of complex homogeneous varieties of high rank. Using the results of \cite{br} one can extend our proofs to show the admissibility of varieties of high rank over local non-archimedian fields. Also using Corollary $4.3$ of \cite{cmpv} one can dispense with the assumption that $X$ is a complete intersection.

math.CO↗
arXiv2017

Appoximate Cohomology

Let $k$ be a field, $G$ be an abelian group and $r\in \mathbb N$. Let $L$ be an infinite dimensional $k$-vector space. For any $m\in End_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty]$ the minimal $R$ such that for any map $A:G \to End_k(L)$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $χ:G\to End_k(L)$ such that $r(A(g)-χ(g))\leq R(G, r, k)$ for all $g\in G$. We show the finiteness of $R(G,r,k)$ for the case when $k$ is a finite field, $G=V$ is a $k$-vector space $V$ of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of {\it Approximate Cohomology} groups $H^k_{\mathcal F} (V,M)$ (which is a purely algebraic analogue of the notion of $ε$-representation (\cite{ep})) and interperate our result as a computation of the group $H^1_{\mathcal F} (V,M)$ for some $V$-modules $M$.

math.GR↗
arXiv2017

Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors

We establish a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform", "uniformly almost periodic", or "nilsequence") behaviour in two different directions separately, then it also exhibits the same behavior (but at higher degree) in both directions jointly. Among other things, this allows one to control averaged local Gowers uniformity norms by global Gowers uniformity norms. In a sequel to this paper, we will apply such control to obtain asymptotics for "polynomial progressions" $n+P_1(r),\dots,n+P_k(r)$ in various sets of integers, such as the prime numbers.

math.CO↗
arXiv2017

On the bias of cubic polynomials

Let $V$ be a vector space over a finite field $k=\mathbb{F} _q$ of dimension $n$. For a polynomial $P:V\to k$ we define the bias of $P$ to be $$b_1(P)=\frac {|\sum _{v\in V}ψ(P(V))|}{q^n}$$ where $ψ:k\to \mathbb{C} ^\star$ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any $d\geq 1$ and $c>0$ there exists $r=r(d,c)$ such that any polynomial $P$ of degree $d$ with $b_1(P)\geq c$ can be written as a sum $P=\sum _{i=1}^rQ_iR_i$ where $Q_i,R_i:V\to k$ are non constant polynomials. We show the validity of a modified version of the converse statement for the case $d=3$.

math.NT↗