Timeline for Number of Laurent monomials of n variables with degree at most d
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 7, 2022 at 22:29 | history | became hot network question | |||
| Jun 7, 2022 at 20:52 | comment | added | Thien | Now, consider the ring of Laurent polynomials $\mathbb R[x,x^{-}]$ with indeterminates $x$ and $x^{-}$. Again, the monimials are the powers of indeterminates, i.e., $1,x,x^2,x^3,\ldots$, and $x^{-1},x^{-2},x^{-3},\ldots$. So, their degrees are also defined by their "powers", e.g., degree of $x^{-3}$ is 3 since $x^{-3}$ is in fact the power 3 of the indeterminates $x^{-}$. So, the definition for the case $n$-variables is adapted thereby. | |
| Jun 7, 2022 at 20:51 | comment | added | Thien | @SamHopkins: Somehow, to me the definition is still adapted with the usual notion. Consider the usual polynomial ring $\mathbb R[x]$ with indeterminate $x$. The usual monimials are the powers of indeterminate, i.e., $1,x,x^2,x^3,\ldots$ Their degrees are the powers as we all known. (Continue) | |
| Jun 7, 2022 at 20:14 | comment | added | Thien | @SamHopkins It's just the definition. Consider Laurent monomials $X=x_1^{a_1}\ldots x_n^{a_n}$, where the exponents $a_1,\dots,a_n$ are integers. Then the degree is defined by $\textrm{deg}(X)=\textrm{max}\{\sum_{a_i>0}a_i, -\sum_{a_i<0}a_i\}$. | |
| Jun 7, 2022 at 19:02 | history | edited | LSpice | CC BY-SA 4.0 |
Clarified from the beginning that these are Laurent monomials
|
| Jun 7, 2022 at 17:57 | vote | accept | Thien | ||
| Jun 7, 2022 at 16:02 | answer | added | David E Speyer | timeline score: 15 | |
| Jun 7, 2022 at 15:06 | comment | added | David E Speyer | The term "crystal ball sequence" is from Conway and Sloane royalsocietypublishing.org/doi/10.1098/rspa.1997.0126 . I believe that in this case it simply is the number of vectors which can be written as the sum of $\leq d$ vectors of the form $e_i - e_j$ for $1 \leq i, j \leq n+1$. | |
| Jun 7, 2022 at 15:01 | comment | added | Michael Lugo | It appears that $|\Delta(n,n)| = \sum_{k=0}^n {n \choose k}^2 {n+k \choose k}$ - I used your formulas to find oeis.org/A005258. Furthermore your sequences $|\Delta(2, d)|, |\Delta(3, d)|, |\Delta(4, d)|$ are in the OEIS as oeis.org/A003215, oeis.org/A005902, and oeis.org/A008384, described as the "crystal ball sequence for A_n lattice" with n = 2, 3, 4. I don't know what the "crystal ball sequence" is but maybe this rings a bell for someone. | |
| Jun 7, 2022 at 15:01 | comment | added | David E Speyer | If you pad your monomials with an extra variable to make the total degree zero, then you are looking at the lattice points in the convex hull of the $\binom{n+1}{2}$ vectors $d(e_i - e_j)$ for $1 \leq i, j \leq n+1$. Some people call this the "$A_n$ root polytope". (Other people use the same name for just the convex hull of $e_i - e_j$ for $1 \leq i < j \leq n+1$.) So keywords worth trying are "root polytope" and "Erhart polynomial", but I didn't find anything quickly. | |
| Jun 7, 2022 at 15:00 | history | edited | Glorfindel | CC BY-SA 4.0 |
added 5 characters in body
|
| S Jun 7, 2022 at 14:29 | review | First questions | |||
| Jun 7, 2022 at 15:00 | |||||
| S Jun 7, 2022 at 14:29 | history | asked | Thien | CC BY-SA 4.0 |