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# Abhinav Kumar

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 Name Abhinav Kumar Member for 3 years Seen 1 hour ago Website Location Age
 2h comment Tensor product with $\mathbb{R}$ of an even unimodular latticeCan you make your question more mathematically precise? 2d comment For what fields is $GL_n(k)$ a rational variety?$GL_n$ is certainly a rational variety (over any field), since it's birational to $M_n$ which is affine $n^2$ space. Jun12 accepted Classification of these Binary Quadratic Forms Jun12 answered Classification of these Binary Quadratic Forms Jun8 comment Formal definitions for a few lattice packing invariantsYes, volume is the volume of the fundamental domain (i.e. the lattice unit cell). Center density is the sphere packing density of the lattice divided by the volume of a unit ball. It's given by $r^n/V$ where $r = \min \Lambda /2$ is the radius of the sphere packing, and $V$ is the volume of the lattice. "Thickness" refers to the thickness of the lattice covering, i.e. the ratio $v R^n/V$, where $v$ is the volume of the unit ball, $R$ the covering radius, and $V$ is the lattice volume as above. Jun7 comment Question about the elementary divisors of a special matrixIn the situation above, I'm thinking of $AB$ acting on $\mathbb{Z}^n$ (column vectors), and we want to understand the cokernel of this matrix. By itself, $B$ has cokernel zero, so we just have to understand the cokernel of $A$. Jun5 comment Reducing a System of Polynomial Equations @Per Alexandersson - sorry, we posted comments almost simultaneously, so I didn't see your comment ... Jun5 comment Reducing a System of Polynomial Equations Compute a Grobner basis? (should be built-in in sage, if not then you can use the command from singular) Jun5 accepted Minimal representation of a polynomial as a linear combination of squares Jun5 comment Minimal representation of a polynomial as a linear combination of squaresAs an example, $x^4 + 2x^3 + 3x^2 + 5x + 7 = (x^2 + x)^2 + 2x^2 + 5x + 7 = (x^2 + x + 1)^2 + 3x + 6$. Think of this as taking square roots near the place $x = \infty$ on the projective line. Jun5 answered Minimal representation of a polynomial as a linear combination of squares Jun4 comment Hodge isometries between K3 surfacesAre you looking for Hodge isometries between the $H^2$ or just between the transcendental lattices? (If the former, any Hodge isometry can be converted to an isomorphism of K3 after composing by a Weyl group element.) Jun4 comment Minimal representation of a polynomial as a linear combination of squaresIf I'm understanding Pourchet's result correctly, it seems to me that one should be able to represent any polynomial $f$ of even degree as a linear combination of six squares. We can assume $f$ is monic, then if you add a sufficiently large constant $c^2$ (with $c$ rational) to $f$, it will become non-negative everywhere: just take $c^2$ larger than the minimum value of $f$, noting $f$ goes to $\infty$ as the argument goes to $\pm \infty$. So by Pourchet, $f + c^2$ is a sum of at most five squares, and $f$ is a linear combination (with $\pm 1$ coeffs) of six squares. I doubt this is optimal! Jun3 answered Tensor product of lattices Jun3 comment Bounds for $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}$@David: Thanks - I'd forgotten that the inverse of $\zeta(s)$ has a nice L-series too! :-) Jun2 revised Question about the elementary divisors of a special matrixadded 11 characters in body Jun2 answered Question about the elementary divisors of a special matrix Jun1 accepted zeros of a homogeneous polynomial Jun1 answered Representation quaternions as matrices May30 answered zeros of a homogeneous polynomial May29 comment Proving that every term of the sequence is an integerThe line breaks are all messed up, but I'm sure you can unentangle them. May29 comment Proving that every term of the sequence is an integer@Barry, you can get the recurrence by doing some linear algebra. Here's some gp code to do the m = 3 line (my indices are shifted by 1, since vectors and matrices in gp start with 1 rather than 0). Then you prove it by induction, of course :-) d = 80; A = matrix(d,d); for(i=1,d,A[i,1] = 1; A[i,2] = 1; A[1,i] = 1); for(i=2,d, for(j=3,d, A[i,j] = (A[i,j-2]*A[i-1,j] + A[i,j-1]*A[i-1,j-1])/A[i-1,j-2])); v = A[4,] m = matrix(40,9,i,j,v[i+j-1]); k = matkerint(m) p = vector(9,i,x^(i-1))*k factor(p[1]) May29 comment A question on Nilpotent MatrixNo worries! MO has its quirks. May29 comment A question on Nilpotent MatrixIt looks correct now. I would suggest appending to your answer rather than editing it completely, so this comment thread makes sense ... May29 comment A question on Nilpotent MatrixThere's a problem with your counterexample: namely that the span of A doesn't just consist of nilpotent matrices ($E_{1,2} + E_{2,1}$ squares to the identity, or to a projection matrix if the dimension is larger than $2$). May29 comment Proving that every term of the sequence is an integerThe degrees of the linear recurrence satisfied by the sequences $(a_{m,n})_n$ are $1,2,4,8,15,26,...$ for $m = 0,1,2,3,4,5,...$, which OEIS recognizes as the Cake numbers (A000125). The characteristic polynomials seem to factor into low-degree pieces. May16 accepted Diophantine equation with primitive nth root of unity May16 comment Diophantine equation with primitive nth root of unity@Mark: Ok, just did! May16 answered Diophantine equation with primitive nth root of unity May15 comment Diophantine equation with primitive nth root of unityThe expression $(- (\xi^k - 1)/(\xi - 1))^n$ is real: it equals the $n$'th power of $\sin(k \pi/n)/\sin(\pi/n)$ times $(-1)^{n+k-1}$, if I calculated correctly. So the thing you're taking absolute value of must be $\pm 1$. If it's $1$, you get the trivial solution $k = an$. If it's $-1$, that tells you an $n$'th root of $\pm 2$ must be in the cyclotomic field, and you can probably work from there to get a contradiction and rule it out. Feb28 comment orthogonal base in unimodular latticeno, that one's an even lattice. so it doesn't have an orthogonal basis. Feb27 accepted orthogonality in a lattice Feb27 revised orthogonality in a latticeedited body Feb27 answered orthogonality in a lattice Feb27 awarded ● Popular Question Feb27 answered orthogonal base in unimodular lattice Feb27 comment How can an integer be factorized as n*m so that n^m has the highest value. This is not really a research-level question, I think. But anyway, a hint is that you're looking to maximize $n^{d/n} = (n^{1/n})^d$. So try to maximize $n^{1/n}$ over divisors of $d$. Feb24 answered Distributing fire stations in a circular city Feb13 comment Order of an element in a finite fieldIf you read the introduction to that paper, they talk about Adleman and DeMarrais's subexponential algorithm for discrete logs in finite fields. The paper itself just describes a probabilistic polynomial time reduction. I don't think anyone knows how to do discrete logs for finite fields in polynomial time (in $\log q$, of course) :-) Feb13 answered Decomposition of primes in Galois closures of number fields Feb12 answered Reference for Complex Abelian Varieties Feb11 accepted Why the Abel-Jacoby map is algebraic morphism? Feb11 answered Efficient algorithm finding ‘a’ solution of system of linear inequalities Feb11 answered Why the Abel-Jacoby map is algebraic morphism? Feb10 answered Solving an equation Feb9 answered To what extent does trajectory determine gravity sources? Feb9 comment To what extent does trajectory determine gravity sources?I think the question is restricting to point masses (as opposed to spread-out regions, like shells). Otherwise one could also use the same argument to replace any point mass by a ball of the same mass (as long as the trajectory does not intersect it). Feb8 comment Sums of two squares: What is known about the distribution of r(n)?The average value is $\pi$, as you can convince yourself by counting the integer points inside a large circle. For more information, see mathworld.wolfram.com/SumofSquaresFunction.html or (say) a book on analytic number theory ... Feb6 accepted linear independence of orbits via a set of transformations in char p Feb6 comment linear independence of orbits via a set of transformations in char pI don't know if there's a principled way to make counterexamples. Keeping in mind what Jason said, you probably need to keep $p$ small. You could try $p = 2$ and $n = 3$, with the non-identity elements of a group isomorphic to $\mathbb{Z}/2 \oplus \mathbb{Z}/2$ (so try to find two idempotent transformations which commute). I don't know if this will work, but it should be easy enough to write a computer program to search. In the other direction, I assume you're already tried rephrasing the problem in a form where you could hit it with Hilbert 90?