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We establish cancellation in short sums of certain special trace functions over $mathbb{F}_q[u]$ below the P{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $mathbb{F}_q[u]$-analog of Chowlas conjecture on cancellation in M{o}bius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $mathbb{F}_q[u]$.
We obtain an estimate for the average value of the product of the Mobius function and any polynomial phase over short intervals and arithmetic progressions simultaneously. As a consequence, we prove that the product of M{o}bius and any polynomial pha
As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${rm PSL}_n(q)$ is prime. We present heuristic
We prove Manins conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.
A vast class of exponential functions is showed to be deterministic. This class includes functions whose exponents are polynomial-like or piece-wise close to polynomials after differentiation. Many of these functions are indeed disjoint from the Mobi
Let $K$ be a local function field of characteristic $l$, $mathbb{F}$ be a finite field over $mathbb{F}_p$ where $l e p$, and $overline{rho}: G_K rightarrow text{GL}_n (mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin metho