Amir Shpilka

Ido Nahshon, Amir Shpilka

We prove that for polynomials $f, g, h \in \mathbb{Z}[x]$ satisfying $f = gh$ and $f(0) \neq 0$, the $\ell_2$-norm of the cofactor $h$ is bounded by $\|h\|_2 \leq \|f\|_1 \cdot\left( \widetilde{O}\left(\|g\|_0^3 \frac{\text{deg }{(f)}^2}{\sqrt{\text{deg }{(g)}}}\right)\right)^{\|g\|_0 - 1}$, where $\|g\|_0$ is the number of nonzero coefficients of $g$ (its sparsity). We also obtain similar results for polynomials over $\mathbb{C}$. This result significantly improves upon previously known exponential bounds (in $\text{deg }{(f)}$) for general polynomials. It further implies that, under exact division, the polynomial division algorithm runs in quasi-linear time with respect to the input size and the number of terms in the quotient $h$. This resolves a long-standing open problem concerning the exact divisibility of sparse polynomials. In particular, our result demonstrates a quadratic separation between the runtime (and representation size) of exact and non-exact divisibility by sparse polynomials. Notably, prior to our work, it was not even known whether the representation size of the quotient polynomial could be bounded by a sub-quadratic function of its number of terms, specifically of $\text{deg }{(f)}$.

Aminadav Chuyoon, Amir Shpilka

We study sparse polynomials with bounded individual degree and their factors, obtaining the following structural and algorithmic results. 1. A deterministic polynomial-time algorithm to find all sparse divisors of a sparse polynomial of bounded individual degree, together with the first upper bound on the number of non-monomial irreducible factors of such polynomials. 2. A $\mathrm{poly}(n,s^{d\log \ell})$-time algorithm that recovers $\ell$ irreducible $s$-sparse polynomials of individual degree at most $d$ from blackbox access to their (not necessarily sparse) product. This partially resolves a question of Dutta-Sinhababu-Thierauf (RANDOM 2024). In particular, if $\ell=O(1)$ the algorithm runs in polynomial time. 3. Deterministic algorithms for factoring a product of $s$-sparse polynomials of individual degree $d$ from blackbox access. Over fields of characteristic zero or sufficiently large characteristic the runtime is $\mathrm{poly}(n,s^{d^3\log n})$; over arbitrary fields it is $\mathrm{poly}(n,(d^2)!,s^{d^5\log n})$. This improves Bhargava-Saraf-Volkovich (JACM 2020), which gives $\mathrm{poly}(n,s^{d^7\log n})$ time for a single sparse polynomial. For a single sparse input we obtain $\mathrm{poly}(n,s^{d^2\log n})$ time. 4. Given blackbox access to a product of factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm to find all irreducible sparse multiquadratic factors with multiplicities. This generalizes the algorithms of Volkovich (RANDOM 2015, 2017) and extends the complete-power test of Bisht-Volkovich (CC 2025). To handle arbitrary fields we introduce a notion of primitive divisors that removes characteristic assumptions from most of our algorithms.

Shai Ben-David, Pavel Hrubes, Shay Moran et al.

We consider the following statistical estimation problem: given a family F of real valued functions over some domain X and an i.i.d. sample drawn from an unknown distribution P over X, find h in F such that the expectation of h w.r.t. P is probably approximately equal to the supremum over expectations on members of F. This Expectation Maximization (EMX) problem captures many well studied learning problems; in fact, it is equivalent to Vapnik's general setting of learning. Surprisingly, we show that the EMX learnability, as well as the learning rates of some basic class F, depend on the cardinality of the continuum and is therefore independent of the set theory ZFC axioms (that are widely accepted as a formalization of the notion of a mathematical proof). We focus on the case where the functions in F are Boolean, which generalizes classification problems. We study the interaction between the statistical sample complexity of F and its combinatorial structure. We introduce a new version of sample compression schemes and show that it characterizes EMX learnability for a wide family of classes. However, we show that for the class of finite subsets of the real line, the existence of such compression schemes is independent of set theory. We conclude that the learnability of that class with respect to the family of probability distributions of countable support is independent of the set theory ZFC axioms. We also explore the existence of a "VC-dimension-like" parameter that captures learnability in this setting. Our results imply that that there exist no "finitary" combinatorial parameter that characterizes EMX learnability in a way similar to the VC-dimension based characterization of binary valued classification problems.

Shay Moran, Amir Shpilka, Avi Wigderson et al.

In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. Namely, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension. Let $C$ be a binary concept class of size $m$ and VC-dimension $d$. Prior to this work, the best known upper bounds for both parameters were $\log(m)$, while the best lower bounds are linear in $d$. We present significantly better upper bounds on both as follows. Set $k = O(d 2^d \log \log |C|)$. We show that there always exists a concept $c$ in $C$ with a teaching set (i.e. a list of $c$-labeled examples uniquely identifying $c$ in $C$) of size $k$. This problem was studied by Kuhlmann (1999). Our construction implies that the recursive teaching (RT) dimension of $C$ is at most $k$ as well. The RT-dimension was suggested by Zilles et al. and Doliwa et al. (2010). The same notion (under the name partial-ID width) was independently studied by Wigderson and Yehudayoff (2013). An upper bound on this parameter that depends only on $d$ is known just for the very simple case $d=1$, and is open even for $d=2$. We also make small progress towards this seemingly modest goal. We further construct sample compression schemes of size $k$ for $C$, with additional information of $k \log(k)$ bits. Roughly speaking, given any list of $C$-labelled examples of arbitrary length, we can retain only $k$ labeled examples in a way that allows to recover the labels of all others examples in the list, using additional $k\log (k)$ information bits. This problem was first suggested by Littlestone and Warmuth (1986).