Ilya Bogdanov

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Name	Ilya Bogdanov
Member for	1 year
Seen	14 hours ago
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Location	Moscow
Age	35

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2d	comment	Periodic tilings of the plane with fundamental domain given by $k$ squares of prescribed size Fo you need a domain to be connected?
Jun 16	comment	A sum-product estimate in Z/p^2Z Since the order of the multiplicative group of ${\mathbb Z}/p^2{\mathbb Z}$ is $p(p-1)$, you should replace $p$ by $p-1$ either in the group order or in the exponent og $g$.I assume the first, since the second is not interesting at all,
Jun 12	comment	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? Yes, I have had this dilemma in mind when I told `blowing up or away'. Sorry for being not that detailed...
Jun 12	accepted	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?
Jun 12	revised	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? added 234 characters in body
Jun 12	revised	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? added 184 characters in body
Jun 12	comment	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? I've added an explanation below the main text. Since I was to type something;), I have also added some words about a generalization.
Jun 12	revised	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? added 774 characters in body; added 36 characters in body
Jun 11	revised	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? added 473 characters in body
Jun 11	answered	Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?
Jun 9	comment	Group with 2 orbits on the nonnegative integers — description of the orbits 1. Yes, I was looking for that, but it seems that the situation is harder here. 2. Yes, to find the density we need much more.
Jun 8	revised	Group with 2 orbits on the nonnegative integers — description of the orbits added 6 characters in body
Jun 8	answered	Group with 2 orbits on the nonnegative integers — description of the orbits
May 29	accepted	Are roots of transcendental elements transcendental?
May 29	comment	Are roots of transcendental elements transcendental? And yes, we may say that $k[t]={\mathbb F}[X,Y]/(Y^2)$, $t=\overline{X}$, and $k$ is a subring generated by $a=\overline{Y}$ and $b=\overline{X}^3\overline{Y}$. Then $at^3-b=0$, but $t^2$ is transcendental.
May 29	comment	Are roots of transcendental elements transcendental? Yes, I was wrong about the term, sorry. The polynomial is simply $at^3-b$...
May 29	revised	Are roots of transcendental elements transcendental? added 3 characters in body; deleted 92 characters in body
May 29	answered	Are roots of transcendental elements transcendental?
Apr 13	comment	“Box Nodes” in Directed Graphs with Paired IO Symmetry As far as I undestand, the question may be formulated as follows. We have a digraph where each vertex has both in- and out-degrees equal to 2. $n$ independent vertices are marked (there are in/outputs of the black box), and for every permutation of the marked vertices there exists an automorphism of the graph inducing exactly this permutation. Is this what you meant?
Apr 9	comment	The sum of same powers of all matrices modulo p The approach is interesting. But it seems that you think there are $p^3$ matrices; in fact, there are $p^{p^2}$ of them. But you do not need so much, since there are less than $p^p$ distinct denominators (all costant terms are ones). This gives the estimate of about $2p^{p+1}$ coefficients to check. Unfortunately, now we have a half of this amount.
Apr 8	awarded	● Nice Answer
Apr 7	revised	The sum of same powers of all matrices modulo p edited body
Apr 7	revised	The sum of same powers of all matrices modulo p deleted 25 characters in body
Apr 7	comment	The sum of same powers of all matrices modulo p Does it remain for larger $k$?
Apr 7	comment	Citation for subset complement result It is so obvious that I do not think it deserves a citation.
Apr 7	answered	The sum of same powers of all matrices modulo p
Apr 5	accepted	Increasing sequence of normal magic squares
Apr 5	revised	Increasing sequence of normal magic squares added 495 characters in body; deleted 8 characters in body
Apr 5	answered	Increasing sequence of normal magic squares
Apr 3	accepted	Lattice basis with Gram-Schmidt vectors of increasing length
Apr 3	answered	Lattice basis with Gram-Schmidt vectors of increasing length
Mar 30	comment	Area of a lattice polygon in terms of its width In fact, this bound is tight. Look at the triangle wth vertices $(0,0)$, $(2,1)$ and $(1,2)$. Its area is $3/2$, while its "integral width" is 2. To see that, first notice that all its altitude lengths are greater than 1, hence it is enough to check only the vectors of length less than 2. This check is straightforward. So, what is this bounty for?
Mar 30	comment	Area of a lattice polygon in terms of its width You need to be a bit more careful, since the affine transforms do not preserve scalar product. So in fact you need to put the vector orthogonal to $v$ into $(-1,1)$.
Mar 28	comment	Area of a lattice polygon in terms of its width Considering the quadrilateral $Q$ of maximal area inscribed into $M$. THen you may pass to the parallelogram with sides parallel to the diagonals of $Q$ --- its area is at most twice the area of $M$. Hence you may restrict yourself to the lattice parallelograms only.
Mar 24	comment	Hobbled rook tour - Hamiltonian cycle on square grid It is a well-known olympiad problem.
Feb 14	accepted	On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients
Feb 14	comment	On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients Well, the construction blows up not the higher terms, but the terms in the middle...
Feb 14	comment	On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients @Gerry: I feel the same...
Feb 14	comment	On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients I have added the way of constructing such an example.
Feb 14	revised	On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients An answer to the auxiliary question added; added 174 characters in body
Feb 13	answered	On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients
Feb 7	comment	Rank of a matrix with missing entries What does "undefined" mean? Do you need the lower bound for all matrices with some prescribed values?
Jan 27	accepted	What is the most extreme set 4 or 5 nontransitive n-sided dice?
Jan 25	comment	What is the most extreme set 4 or 5 nontransitive n-sided dice? Both numbers 6 and 8 are optimal.
Jan 25	revised	What is the most extreme set 4 or 5 nontransitive n-sided dice? added 316 characters in body
Jan 25	answered	What is the most extreme set 4 or 5 nontransitive n-sided dice?
Jan 25	accepted	A quantitative version of Straszewicz’s theorem?
Jan 25	answered	A quantitative version of Straszewicz’s theorem?
Jan 5	comment	Covering all, but $k$ points with affine subspaces @domotorp: By a straightforward induction on $p$, the subset uncovered by $p$ hyperplanes, if nonempty, is an affine subspace of codimension at most $p$.
Jan 4	accepted	Covering of a partial order by upwards convex sets

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