Let $X\subset {\mathbb{P}}^n$ be an integral and non-degenerate $m$-dimensional variety defined over $ℝ$. For any $P\in {\mathbb{P}}^n\left(ℝ\right)$ the real $X$-rank $r_{X,ℝ}\left(P\right)$ is the minimal cardinality of $S\subset X\left(ℝ\right)$ such that $P\in \langle S\rangle$. Here we extend to the real case an upper bound for the $X$-rank due to Landsberg and Teitler.

Let $K$ be a field of odd characteristic $p$, let $f\left(x\right)$ be an irreducible separable polynomial of degree $n\ge 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^2=f\left(x\right)$ and $J\left(C\right)$ its jacobian. We prove that $J\left(C\right)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n\ge 7$ or $p\ne 3$.

Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

Let $G$ be a semisimple linear algebraic group of inner type over a field $F$, and let $X$ be a projective homogeneous $G$-variety such that $G$ splits over the function field of $X$. We introduce the $J$-invariant of $G$ which characterizes the motivic behavior of $X$, and generalizes the $J$-invariant defined by A. Vishik in the context of quadratic forms. We use this $J$-invariant to provide motivic decompositions of all generically split projective homogeneous $G$-varieties, e.g. Severi-Brauer varieties,...

Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q\left(m\right)$, where $Q$ is a polynomial with integral coefficients, then $P\left(x\right)=Q\left(R\right(x\left)\right)$ for some polynomial $R$. In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^q$ where $q$ is an integer greater than 1, then $P\left(x\right)={\left(R\left(x\right)\right)}^q$ for some polynomial $R\left(x\right)$.

We consider tangent cones to Schubert subvarieties of the flag variety $G/B$, where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_6$, $E_7$ or $E_8$. We prove that if $w_1$ and $w_2$ form a good pair of involutions in the Weyl group $W$ of $G$ then the tangent cones $C_{w_1}$ and $C_{w_2}$ to the corresponding Schubert subvarieties of $G/B$ do not coincide as subschemes of the tangent space to $G/B$ at the neutral point.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $\mathrm{pos}\le c$ positive and $\neg \le p$ negative roots, where $\mathrm{pos}\equiv c(mod2)$ and $\neg \equiv p(mod2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(\mathrm{pos},\mathrm{neg})$ satisfying these conditions there exists a polynomial $P$ with exactly $\mathrm{pos}$ positive and exactly $\neg$ negative roots (all of them simple). For $d\ge 4$...

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Let $𝒱$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C\left(𝒱\right)$ such that for almost all primes $p$ for all but at most $C\left(𝒱\right)$ points on the reduction of $𝒱$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

For an integer $k\ge 2$, let ${\left(n\right)}_n$ be the $k-$generalized Pell sequence which starts with $0,...,0,1$ ($k$ terms) and each term afterwards is given by the linear recurrence $n=2n-1+n-2+\cdots +n-k$. In this paper, we find all $k$-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${\left(P_n^{\left(2\right)}\right)}_n$. ...

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$, and let $\overline{X}$ and $\overline{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br $\left(\overline{X}\right)\oplus$ Br($\overline{Y})$ has finite index in the Galois invariant subgroup of Br$(\overline{X}\times \overline{Y})$. This implies that the cokernel of the natural map Br$\left(X\right)\oplus$ Br$\left(Y\right)\to$ Br$(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the...