arXiv subjects

Amichai Lampert

Publications and source records attributed to Amichai Lampert.

13 recordsLinked to original sources

arXiv2026

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↗

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

Slice rank and analytic rank for trilinear forms

In this note, we present an elementary proof of the fact that the slice rank of a trilinear form over a finite field is bounded above by a linear expression in the analytic rank. The existing proofs by Adiprasito-Kazhdan-Ziegler and Cohen-Moshkovitz both rely on results of Derksen via geometric invariant theory. A novel feature of our proof is that the linear forms appearing in the slice rank decomposition are obtained from the trilinear form by fixing coordinates.

math.CO↗

arXiv2025

Slice rank and partition rank of the determinant

The Laplace expansion expresses the $n \times n$ determinant $\det_n$ as a sum of $n$ products. Do shorter expansions exist? In this paper we: - Fully determine the slice rank decompositions of $\det_n$ (where each product must contain a linear factor): In this case, we show that $n$ summands are necessary, and moreover, the only such expansions with $n$ summands are equivalent (in a precise sense) to the Laplace expansion. - Prove a logarithmic lower bound for the partition rank of $\det_n$ (where each product is of multilinear forms): In this case, we show that at least $\log_2(n)+1$ summands are needed. We also explain why existing techniques fail to yield any nontrivial lower bound, and why our new method cannot give a super-logarithmic lower bound. - Separate partition rank from slice rank for $\det_n$: we find a quadratic expansion for $\det_4$, over any field, with fewer summands than the Laplace expansion. This construction is related to a well-known example of Green-Tao and Lovett-Meshulam-Samorodnitsky disproving the naive version of the Gowers Inverse conjecture over small fields. An important motivation for these questions comes from the challenge of separating structure and randomness for tensors. On the one hand, we show that the random construction fails to separate: for a random tensor of partition rank $r$, the analytic rank is $r-o(1)$ with high probability. On the other hand, our results imply that the determinant yields the first asymptotic separation between partition rank and analytic rank of $d$-tensors, with their ratio tending to infinity with $d$.

math.CO↗

arXiv2025

Strength is bounded linearly by Birch rank

Let $f$ be a homogeneous polynomial over a field. For many fields, including number fields and function fields, we prove that the strength of $f$ is bounded above by a constant multiple of the Birch rank of $f.$ The constant depends only on the degree of $f$ and the absolute transcendence degree of the field. This is the first linear bound obtained for forms of degree greater than three, partially resolving a conjecture of Adiprasito, Kazhdan and Ziegler. Our result has applications for the Hardy-Littlewood circle method. The circle method yields an asymptotic formula for counting integral zeros of (collections of) homogeneous polynomials, provided the Birch rank is sufficiently large -- a natural geometric condition. Our main theorem implies that these formulas hold even if we only assume a similar lower bound on the strength of the (collection of) homogeneous polynomials -- an arithmetic condition which is a priori weaker. This answers questions of Cook-Magyar and Skinner, and also yields a new proof of a seminal result of Schmidt as a consequence of Birch's earlier work. Over finite fields we obtain a quasi-linear bound for partition rank of tensors in terms of analytic rank, improving Moshkovitz-Zhu's state of the art bound.

math.NT↗

arXiv2025

Density of solutions for systems of forms

Let $K$ be a field of characteristic zero over which every diagonal form in sufficiently many variables admits a nontrivial solution. For example, $K$ may be a totally imaginary number field or a finite extension of a $p$-adic field. Suppose $f_1,\ldots,f_s$ are forms of degree $d$ over $K.$ Bik, Draisma and Snowden recently proved that there exists a constant $B = B(d,s,K)$ such that the rational solutions to the system of equations $f_1=\ldots=f_s = 0$ are Zariski dense, as long as the Birch rank of $f_1,\ldots,f_s$ is greater than $B.$ We establish an effective bound for this constant, improving vastly on the astronomical bound coming from their proof. Our result has applications for surjectivity of polynomial maps and for the Hardy-Littlewood circle method.

math.NT↗

arXiv2024

Two improvements in Birch's theorem on forms

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the field of rational numbers. Let $f_1, \ldots, f_r$ be homogeneous forms of odd degree over $K$ in $n$ variables, and let $Z$ be the variety they cut out. Birch proved if $n$ is sufficiently large then $Z(K)$ contains a non-zero point. We prove two results which show that $Z(K)$ is actually quite large. First, the Zariski closure of $Z(K)$ has bounded codimension in $\mathbf{A}^n$. And second, if the $f_i$'s have sufficiently high strength then $Z(K)$ is in fact Zariski dense in $Z$. The proofs use recent results on strength, and our methods build on recent work of Bik, Draisma, and Snowden, which established similar improvements to Brauer's theorem on forms.

math.NT↗

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

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↗

arXiv2023

Small ideals in polynomial rings and applications

Let $\mathbf{k}$ be a field which is either finite or algebraically closed and let $R = \mathbf{k}[x_1,\ldots,x_n].$ We prove that any $g_1,\ldots,g_s\in R$ homogeneous of positive degrees $\le d$ are contained in an ideal generated by an $R_t$-sequence of $\le A(d)(s+t)^{B(d)}$ homogeneous polynomials of degree $\le d,$ subject to some restrictions on the characteristic of $\mathbf{k}.$ This yields effective bounds for new cases of Ananyan and Hochster's theorem A in arXiv:1610.09268 on strength and the codimension of the singular locus. It also implies effective bounds when $d$ equals the characteristic of $\mathbf{k}$ for Tao and Ziegler's result in arXiv:1101.1469 on rank and $U^d$ Gowers norms of polynomials over finite fields.

math.AC↗

arXiv2023

Schmidt rank and singularities

We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous polynomials of degree d (assuming that the characteristic does not divide d(d-1)). We then use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces). This gives an effective version of Ananyan-Hochster's Theorem A from arXiv:1610.09268.

math.AG↗

arXiv2019

Bias implies low rank for quartic polynomials

We investigate the structure of polynomials of degree four in many variables over a fixed prime field $\mathbb{F}=\mathbb{F}_{p}$. In 2007, Green and Tao proved that if a polynomial $f:\mathbb{F}^{n}\rightarrow\mathbb{F}$ is poorly distributed, then it is a function of a few polynomials of smaller degree. In 2009, Haramaty and Shpilka found an effective bound for $f$ of degree four: If $bias\left(f\right)\geqδ$, then the number of lower degree polynomials required is at most polynomial in $1/δ$ and $f$ has a simple presentation as a sum of their products. We make a step towards showing that in fact the number of lower degree polynomials required is at most log-polynomial in $1/δ$, with the same simple presentation of $f$. This result was a Master's thesis supervised by T. Ziegler at the Hebrew University of Jerusalem, submitted in October 2018. A log-polynomial bound for polynomials of arbitrary degree was recently proved independently by Milicevic and by Janzer.

math.CO↗

arXiv2012

On Maximizing the Speed of a Random Walk in Fixed Environments

We consider a random walk in a fixed Z environment composed of two point types: (q,1-q) and (p,1-p) for 1/2<q<p. We study the expected hitting time at N for a given number k of p-drifts in the interval [1,N-1], and find that this time is minimized asymptotically by equally spaced p-drifts.

math.PR↗