Smooth polynomials with several prescribed coefficients

Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_q$ of $q$ elements. A polynomial in $\mathbb{F}_q[t]$ is called $m$-smooth (or $m$-friable) if all its irreducible factors are of degree at most $m$. In this paper, we investigate the distribution of $m$-smooth (or $m$-friable) polynomials with prescribed coefficients. Our technique is based on character sum estimates on smooth (friable) polynomials proved by the analytic approach of Panario, Gourdon, Flajolet (1998), Bourgains's argument (2015) applied for polynomials by Ha (2016) and on double character sums on smooth (friable) polynomials

Development and application of functional composite surfaces with tunable wettability and photoreactivity

The popularity of functional surfaces with either extreme wetting properties or photocatalytic properties is rapidly increasing. During my doctoral work, I prepared composite surfaces with both features, applying visible light-active, plasmonic Ag-TiO2 nanoparticles, polyacrylates and PDMS. At first, I presented that the hydrophobic wetting character of perfluorinated polyacrylate films was enhanced as a result of surface roughening with Ag-TiO2. The prepared coatings possessed Ag-TiO2/FP-ratio- and roughness-dependent wettability (and photoreactivity), ranging from superhydrophilic to superhydrophobic character. Besides keeping the 80 wt.% photocatalyst content, the addition of a hydrophilic matrix component poly(2-hydroxyethyl acrylate, or pHEA) could also lead to superhydrophilic composites. As the popularity of microfluidics and liquid manipulation is increasing, the need for the development of novel, stimulus-responsive surfaces with tunable wettability also emerges. In alignment with this, I also prepared composites on PDMS basis with magneto- and thermoresponsive wetting character and photoreactivity, applying carbonyl iron microparticles and poly(N-isopropyl acrylamide). The main hindering factor during the practical application of surfaces with extreme wettability is the vulnerability of the surface fine structure: to surpass this disadvantage and to achieve lasting superhydrophobicity, PDMS oleogel-based self-healing composites were also prepared. Due to their multifunctional nature, the presented composite surfaces may will be applicable for the decomposition of pollutants with varying polarity or for the implementation of complex liquid manipulation tasks