arXiv subjects

Eli Amzallag

Publications and source records attributed to Eli Amzallag.

3 recordsLinked to original sources

arXiv2021

Degree bound for toric envelope of a linear algebraic group

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of the degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group $G \subset \mathrm{GL}_n(C)$ can be arbitrarily large even for $n = 1$. One of the key ingredients of Hrushovski's algorithm for computing the Galois group of a linear differential equation was an idea to `approximate' every algebraic subgroup of $\mathrm{GL}_n(C)$ by a `similar' group so that the degree of the latter is bounded uniformly in $n$. Making this uniform bound computationally feasible is crucial for making the algorithm practical. In this paper, we derive a single-exponential degree bound for such an approximation (we call it toric envelope), which is qualitatively optimal. As an application, we improve the quintuply exponential bound for the first step of the Hrushovski's algorithm due to Feng to a single-exponential bound. For the cases $n = 2, 3$ often arising in practice, we further refine our general bound.

math.AG↗

arXiv2021

Evidence of Random Matrix Corrections for the Large Deviations of Selberg's Central Limit Theorem

Selberg's central limit theorem states that the values of $\log|ζ(1/2+i τ)|$, where $τ$ is a uniform random variable on $[T,2T]$, is distributed like a Gaussian random variable of mean $0$ and standard deviation $\sqrt{\frac{1}{2}\log \log T}$. It was conjectured by Radziwiłł that this breaks down for values of order $\log\log T$, where a multiplicative correction $C_k$ would be present at level $k\log\log T$, $k>0$. This constant should be equal to the leading asymptotic for the $2k^{th}$ moment of $ζ$, as first conjectured by Keating and Snaith using random matrix theory. In this paper, we provide numerical and theoretical evidence for this conjecture. We propose that this correction has a significant effect on the distribution of the maximum of $\log|ζ|$ in intervals of size $(\log T)^θ$, $θ>0$. The precision of the prediction enables the numerical detection of $C_k$ even for low $T$'s of order $T=10^8$. A similar correction appears in the large deviations of the Keating-Snaith central limit theorem for the logarithm of the characteristic polynomial of a random unitary matrix, as first proved by Féray, Méliot and Nikeghbali.

math.PR↗

arXiv2018

Complexity of Triangular Representations of Algebraic Sets

Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szanto. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds.

math.AG↗