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Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb F_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani'c conjectured that $C_k(q)=1+q+\cdots+q^k$. We disprove the unrestricted conjecture by constructing thirteen monic poly...
For a prime \(p\), let \(A_p=\{k!\pmod p:1\leq k<p\}\). We prove \(|A_p|\gg p^{8/15}\), improving the general lower bound \((\sqrt{2}-o(1))p^{1/2}\). The proof begins with the identity \((n+2)!=(n+1)!+((n+1)!)^2/n!\) in \(\mathbb{F}_p\), which produces many incidences for a family of fractional-linear maps. After Cauch...
For an integer $d\ge 3$, put $\Delta_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\De...
We develop two transition principles for lower-bounding value sets generated by structured sequences over prime fields. A reciprocal-affine family with $M$ internal transitions and bounded quotient multiplicity has image size $\gg \min{M,p}^{8/15}$. This recovers the factorial-residue bound and yields the same exponent...
Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codime...
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