User oleg eroshkin - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:18:36Z http://mathoverflow.net/feeds/user/1811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/133415/lattice-points-in-cross-polytopes Lattice points in cross-polytopes Oleg Eroshkin 2013-06-11T18:58:33Z 2013-06-15T22:18:38Z <p>Let $E\subset \mathbb{R}^n$ be a cross-polytope: $$E= \left\lbrace x : \frac{|x_1|}{q_1}+\cdots+\frac{|x_n|}{q_n}\leq 1 \right\rbrace,$$ where $q_1,\dots,q_n$ are positive integers. I am interested in estimating the number $L$ of lattice points in $E$.</p> <p><strong>Conjecture.</strong> <code>$$L:=\# \left(E \cap \mathbb{Z}^n\right)&gt;\mathrm{vol}(E)\,.$$</code></p> <p>It's obviously true for $n=1$, and very easy to verify (using Pick's formula) for $n=2$. I don't know how to prove it for $n\geq 3$ without some additional constraints.</p> <p>According to (generalized) Blichfeldt's theorem, there exists $x\in\mathbb{R}^n$ such that $E+x$ has more lattice points than $\mathrm{vol}(E)$. This suggests the following</p> <p><strong>Conjecture (strong version).</strong> For any $x\in\mathbb{R}^n$ <code>$$L\geq \#\left((E+x)\cap\mathbb{Z}^n\right)\,.$$</code></p> <p>I don't know how to prove (or disprove) this version even for $n=2$.</p> http://mathoverflow.net/questions/128951/reference-request-samuels-multiplicity-and-degree Reference request: Samuel's multiplicity and degree Oleg Eroshkin 2013-04-27T20:07:18Z 2013-06-01T21:13:44Z <p>I am looking for references for the following simple facts.</p> <ol> <li><p>Let $Y\subset \mathbb{P}^n$ be a variety (or pure-dimensional algebraic set). For $P\in Y$ denote by $e_p(Y)$ the (Samuel's) multiplicity of $Y$ at $P$. Then $\deg Y\geq e_p(Y)$.</p></li> <li><p>For simplicity, I will assume that $Y\subset K^n$ is affine variety. Suppose that for any $f\in I(Y)$ derivatives $\partial^a f(P)=0$ for all multi-indices $a=(a_1,\dots,a_n)$ satisfying $a_1/w_1+\cdots+a_n/w_n&lt;1$. Then $e_p(Y)\geq \min w_{i_1} w_{i_2}\dots w_{i_d}$, where $d=\dim Y$.</p></li> </ol> <p>I assume that the $K$ is algebraically closed field of characteristic 0.</p> <p>I know how to prove these facts. For example, the first claim follows immediately from the Corollary 12.4 in Fulton's Intersection theory.</p> <p><em>Added later:</em> I was surprised to find a nice geometric description of the difference $\deg Y - e_p(Y)$ in the Appendix to Chapter 6 in Mumford's "Algebraic Geometry I: Complex Projective Varieties". I completely forgot about it.</p> <p>The part 2 is <em>false</em> without some additional conditions on $Y$ (Cohen-Macaulay is sufficient).</p> http://mathoverflow.net/questions/64643/height-of-algebraic-numbers/64669#64669 Answer by Oleg Eroshkin for Height of algebraic numbers Oleg Eroshkin 2011-05-11T18:14:04Z 2011-05-11T18:35:26Z <p>If you know only heights of $a$ and $b$, you may estimate heights of $a+b$, $a/b$ and $ab$. Assuming that $h$ is an absolute (Weil) height: $$h(ab)\leq h(a)+h(b)$$ $$h(a/b)\leq h(a)+h(b)$$ $$h(a+b)\leq\log 2 +h(a)+h(b)$$ This bounds are sharp. You may find this, for example, in M. Waldschmidt "Diophantine approximation on linear algebraic groups", Chapter 3.</p> http://mathoverflow.net/questions/41011/what-is-the-indefinite-sum-of-tanx/41098#41098 Answer by Oleg Eroshkin for What is the indefinite sum of tan(x)? Oleg Eroshkin 2010-10-05T02:38:21Z 2010-10-17T18:09:07Z <p>There are no "nice" functions with such properties. Every solution is discontinuous at a dense subset of $\mathbb{R}$. Just look on poles of $\tan x$. Let $x=\pi/2+\pi m$, $m\in\mathbb{Z}$. Clearly either $T$ is discontinuous at $x$ or at $x+1$. In the latter case it is also discontinuous at $x+k$ for every positive integer $k$. In the former case, $T$ is discontinuous at $x+k$ for every non-positive integer $k$.</p> http://mathoverflow.net/questions/40828/obstruction-for-real-subvariety-to-be-embedded-as-complex-subvariety/41037#41037 Answer by Oleg Eroshkin for Obstruction for real subvariety to be embedded as complex subvariety Oleg Eroshkin 2010-10-04T16:20:38Z 2010-10-04T16:20:38Z <p>Actually answer is no even when you allow real embedding of $\mathbb{CP}^n$. Consider any null-homotopic embedding of $X$ into $\mathbb{CP}^n$. Such embedding cannot be a pullback of a complex submanifold of $\mathbb{CP}^N$.</p> http://mathoverflow.net/questions/40046/equidistant-points-in-negatively-curved-metric-spaces/40047#40047 Answer by Oleg Eroshkin for Equidistant points in negatively curved metric spaces Oleg Eroshkin 2010-09-26T20:10:23Z 2010-09-26T20:10:23Z <p>Hello Dave,</p> <p>Three disks of equal radius in Euclidean plane with centers on a circle of sufficiently large radius seems to be an easy counter-example.</p> http://mathoverflow.net/questions/7732/diameter-of-m-fold-cover/8357#8357 Answer by Oleg Eroshkin for Diameter of m-fold cover Oleg Eroshkin 2009-12-09T15:12:33Z 2009-12-09T15:12:33Z <p>I can show that $diam(\widetilde{M})\leq (m+1)diam(M)$. It follows from the fact that the fundamental group of $M$ is generated by "short" loops of length at most $2diam(M)$ (this is proved in Gromov's book "Metric structures ..."). Lets show that if $p$ and $q$ in $\widetilde{M}$ have the same projection $x$ in $M$, then $dist(p,q)\leq [m/2]\cdot diam(M)$. Consider the following graph. Vertices are $m$ preimages of $x$, edges correspond to short loops in $M$. This graph is 2-connected, therefore the diameter of the graph is at most $[m/2]$. </p> <p>To improve the estimate it is sufficient to show, that for every two points $x,y\in M$ short loops with based point $x$ through $y$ generates the fundamental group.</p> http://mathoverflow.net/questions/7594/when-is-a-cx-cy-a-composition-operator/7812#7812 Answer by Oleg Eroshkin for When is A : C(X) --> C(Y) a composition operator? Oleg Eroshkin 2009-12-04T20:44:51Z 2009-12-04T20:44:51Z <p>For hemicompact k-space $X$ the space of continuous homomorphisms of algebra $C(X)$ to &#8450; is $X$ (up to the obvious isomorphism). The proof can be found, for example, in H. Goldmann "Uniform Frechet Algebras". Then the same construction as for compact spaces give you the map $T$.</p> http://mathoverflow.net/questions/3965/minimal-surface-in-a-ball/5603#5603 Answer by Oleg Eroshkin for Minimal surface in a ball Oleg Eroshkin 2009-11-15T03:11:52Z 2009-11-15T03:11:52Z <p>This result (and several similar) proved in a nice paper Alexander, H.; Osserman, R. "Area bounds for various classes of surfaces." Amer. J. Math. 97 (1975), no. 3, 753--769. </p> http://mathoverflow.net/questions/133415/lattice-points-in-cross-polytopes/133422#133422 Comment by Oleg Eroshkin Oleg Eroshkin 2013-06-11T20:09:45Z 2013-06-11T20:09:45Z Great! That is quite surprising. I assumed, that is better to look for co-prime $q_i$ for counterexamples. http://mathoverflow.net/questions/64709/diophantine-approximation/64713#64713 Comment by Oleg Eroshkin Oleg Eroshkin 2011-05-12T00:48:05Z 2011-05-12T00:48:05Z @Felipe Good point. Than even this conjecture (wide open) is too weak for vanvu. http://mathoverflow.net/questions/64709/diophantine-approximation/64713#64713 Comment by Oleg Eroshkin Oleg Eroshkin 2011-05-12T00:28:38Z 2011-05-12T00:28:38Z Roth's theorem gives bound for $|q\alpha-p|$. This is the linear polynomial. Vanvu need such bounds for polynomials of degree $d$. Conjecturally for algebraic number $\alpha$ and $\epsilon&gt;0$ there are only finitely many polynomials $P$ of degree $d$ with integer coefficients of absolute values at most $N$ with $0&lt;|P(\alpha)|&lt;N^{-d-1-\epsilon}$. http://mathoverflow.net/questions/64709/diophantine-approximation/64713#64713 Comment by Oleg Eroshkin Oleg Eroshkin 2011-05-12T00:09:06Z 2011-05-12T00:09:06Z This is Liouville's inequality. For $d=1$, Roth's theorem gives required bounds. The generalization of Roth theorem for polynomials of higher degree is open. I doubt that the special case of $\alpha$ is significantly easier. http://mathoverflow.net/questions/40828/obstruction-for-real-subvariety-to-be-embedded-as-complex-subvariety Comment by Oleg Eroshkin Oleg Eroshkin 2010-10-02T15:29:07Z 2010-10-02T15:29:07Z Colin, do you mean to embed $\mathbb{CP}^n$ as a real submanifold? http://mathoverflow.net/questions/27573/rational-numbers-with-dense-orbits-in-0-1-under-iteration-by-fx4x1-x/27580#27580 Comment by Oleg Eroshkin Oleg Eroshkin 2010-06-09T15:29:19Z 2010-06-09T15:29:19Z Greg, clearly this condition is the same as normality. Just add different prefixes to the string. http://mathoverflow.net/questions/7594/when-is-a-cx-cy-a-composition-operator/7812#7812 Comment by Oleg Eroshkin Oleg Eroshkin 2009-12-05T01:29:50Z 2009-12-05T01:29:50Z I don't remember if there are examples of hemicompact spaces (without k-space propery) such that $M(C(X))\neq X$ (I don't have Goldmann's book at home). Here, $M(C(X))$ is the space of all continuous homomorphisms of the algebra $C(X)$ to &amp;#8450;. It seems, that $C(M(C(X)))=C(X)$ (is it?). Then try $Y=M(C(X))$ and identity maps between $C(X)$ and $C(Y)$.