Let $G$ be a reductive complex algebraic group, and let $C[mV{]}^G$ denote the algebra of invariant polynomial functions on the direct sum of $m$ copies of the representations space $V$ of $G$. There is a smallest integer $n=n\left(V\right)$ such that generators and relations of $C[mV{]}^G$ can be obtained from those of $C[nV{]}^G$ by polarization and restitution for all $m\>n$. We bound and the degrees of generators and relations of $C[nV{]}^G$, extending results of Vust. We apply our techniques to compute the invariant theory of binary cubics. ...

For a graph G and even integers b ⩾ a ⩾ 2, a spanning subgraph F of G such that a ⩽ degF (x) ⩽ b and degF (x) is even for all x ∈ V (F) is called an even [a, b]-factor of G. In this paper, we show that a 2-edge-connected graph G of order n has an even [2, b]-factor if [...] max degG (x),degG (y)⩾max 2n2+b,3 $max\{{deg}_G\left(x\right),{deg}_G\left(y\right)\}\ge max\left\{\frac{2n}{2+b},3\right\}$ for any nonadjacent vertices x and y of G. Moreover, we show that for b ⩾ 3a and a > 2, there exists an infinite family of 2-edge-connected graphs G of order n with δ(G) ⩾ a...

Sia $X$ una curva liscia di genere $g\ge 2$ ed $A$, $B$ fasci coerenti su $X$. Sia ${\mu }_{A,B}:H^0\left(X,A\right)\otimes H^0\left(X,B\right)\to H^0\left(X,A\otimes B\right)$ l'applicazione di moltiplicazione. Qui si dimostra che ${\mu }_{A,B}$ ha rango massimo se $A\cong {\omega }_X$ e $B$ è un fibrato stabile generico su $X$. Diamo un'interpretazione geometrica dell'eventuale non-surgettività di ${\mu }_{A,B}$ quando $A,B$ sono fibrati in rette generati da sezioni globali e $\mathrm{deg}\left(A\right)+\mathrm{deg}\left(B\right)\ge 3g-1$. Studiamo anche il caso $\text{dim}\left(\text{Coker}\left({\mu }_{A,B}\right)\right)\ge 2$.

Let $K$ be a finite field and $Q\in K\left[T\right]$ a polynomial of positive degree. A function $f$ on $K\left[T\right]$ is called (completely) $Q$-additive if $f(A+BQ)=f\left(A\right)+f\left(B\right)$, where $A,B\in K\left[T\right]$ and $deg\left(A\right)\<deg\left(Q\right)$. We prove that the values $(f_1\left(A\right),...,f_d\left(A\right))$ are asymptotically equidistributed on the (finite) image set $\left\{\right(f_1\left(A\right),...,$ $f_d\left(A\right)):A\in K\left[T\right]\}$ if $Q_j$ are pairwise coprime and $f_j:K\left[T\right]\to K\left[T\right]$ are $Q_j$-additive. Furthermore, it is shown that $(g_1\left(A\right),g_2\left(A\right))$ are asymptotically independent and Gaussian if $g_1,g_2:K\left[T\right]\to ℝ$ are $Q_1$- resp. $Q_2$-additive.

Let $X\to B$ an elliptic fibration with general fibre $F$. Let $n_e,n_s,n_a,n_v$ be the minima of the non-zero intersection numbers $(ℒ,F)$ where $ℒ$ runs successively through the following sets: effective divisors on $X$, invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let $m$ be the maximum of the multiplicities of the fibres of $X\to B$. We prove that $n_e=n_s$ if and only if $n_e\ge 2m$ and that $n_a=n_v$ if and only if $n_a\ge 3m$.

Agrawal, Kayal, and Saxena recently introduced a new method of proving that an integer is prime. The speed of the Agrawal-Kayal-Saxena method depends on proven lower bounds for the size of the multiplicative semigroup generated by several polynomials modulo another polynomial $h$. Voloch pointed out an application of the Stothers-Mason ABC theorem in this context: under mild assumptions, distinct polynomials $A,B,C$ of degree at most $1.2degh-0.2degradABC$ cannot all be congruent modulo $h$. This paper presents two...