Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }{X}_{+}$, where $X_{+}=X??$, for a smooth scheme $X\in {\mathrm{Sm}}_k$ over an infinite perfect field k, is computed.The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $({\mathbb{A}}^l\times X)/(({\mathbb{A}}^l?0)\times X)$, $X\in {\mathrm{Sm}}_k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety X over k and an open subscheme $U?X$.The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }(X_{+}/U_{+})$ of the quotient-sheaf $X_{+}/U_{+}$.     

On discrete Brunn-Minkowski and isoperimetric type inequalities

We show that the lattice point enumerator ${\mathrm{G}}_n(?)$ satisfies${\mathrm{G}}_n{(tK+sL+{(?1,?t+s?)}^n)}^{1/n}\ge t{\mathrm{G}}_n{(K)}^{1/n}+s{\mathrm{G}}_n{(L)}^{1/n}$ for any $K,L?{\mathbb{R}}^n$ bounded sets with integer points and all $t,s\ge 0$.We also prove that a certain family of compact sets, extending that of cubes ${[?m,m]}^n$, with $m\in \mathbb{N}$, minimizes the functional ${\mathrm{G}}_n(K+t{[?1,1]}^n)$, for any $t\ge 0$, among those bounded sets $K?{\mathbb{R}}^n$ with given positive lattice point enumerator.Finally, we show that these new discrete inequalities imply the corresponding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.     

Permutation trinomials over Fq3

We consider four classes of polynomials over the fields ${\mathbb{F}}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+A{x}^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+A{x}^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+A{x}^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+A{x}^q-Bx$, where $A,B\in {\mathbb{F}}_q$. We find sufficient conditions on the pairs $(A,B)$ for which these polynomials permute ${\mathbb{F}}_{q^3}$ and we give lower bounds on the number of such pairs.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.