Timeline for beyond differentially algebraic power series
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 5, 2010 at 5:12 | comment | added | Wadim Zudilin | In fact, I posted a new question about the functional equations. I wonder whether this problem was ever studied. | |
| May 5, 2010 at 5:03 | comment | added | Martin Rubey | The equation for the truncated j (with q^-1 removed) is much larger... (coefficient degree 5, degree in x and degree in y equal to 3). I was too supid to simply try all coefficients! What would you like to investigate with respect to the functional equations? | |
| May 4, 2010 at 21:51 | comment | added | Wadim Zudilin | Yes, modular polynomials are large. For example, $\Phi_2(j(q)j(q^2))=0$ for $\Phi_2(x,y)=-(xy)^2+x^3+y^3+2^4\cdot3\cdot31xy(x+y)+3^4\cdot5^3\cdot4027xy-2^4\cdot3^4\cdot5^3(x^2+y^2)+2^8\cdot3^7\cdot5^6(x+y)-2^{12}\cdot3^9\cdot5^9$. | |
| May 4, 2010 at 17:10 | comment | added | Martin Rubey | Oh, I think I found it, but it's quite large. (I took the q-expansion of j and omitted the coefficient of q^-1) | |
| May 4, 2010 at 16:59 | comment | added | Martin Rubey | Wadim, I couldn't find the functional equation for j. Could you please expand once more... | |
| May 3, 2010 at 23:11 | comment | added | Wadim Zudilin | And yes to your last question: mixing differentiation and substitution $q\mapsto q^k$ is exactly what I mean. | |
| May 3, 2010 at 23:09 | comment | added | Wadim Zudilin | Hi Martin, the $F_k$'s are presumably not tied by any (nontrivial) algebraic relations, that's why I feel them boring enough. All the monomials in $F_k$'s I suggest are linearly independent. (But weren't you interested in having no algebraic dependencies?!) The Eisenstein series example is phantastic because of possessing both the differential and Mahler-type structure. And the latter is, to my knowledge, not investigated... The functional equations are done for the modular invariant $j$ only (mathoverflow.net/questions/22001). | |
| May 3, 2010 at 16:08 | comment | added | Martin Rubey | Wadim, I just checked that E2, E4 and E6 indeed satisfy what I called a Mahler-type functional equation (no idea whether this is good terminology). Thanks for pointing that out - I am surprised! (I did know that they satisfy ADEs, however...) However, I couldn't find any relation for the F_k. Are you saying that they satisfy a "mixed" equation? If so, do you happen to know it? (my program would require some tweaking...) (I guess you really meant I should try to mix differentiation and substitution q +-> q^k) | |
| Apr 30, 2010 at 12:10 | comment | added | Wadim Zudilin | In the case of $F_{2m+1}(q)$ I am probably to speculative in adding $D_k$'s: already $D$ gives a rich structure. If we assign weight $2k+2m+1$ to each $D^kF_{2m+1}(q)$, then the set of monomials $D^{k_1}F_{2m_1+1}\dots D^{k_s}F_{2m_s+1}$ of given weight is finite, so one may introduce their ordering. In your example you have a single function which does not satisfy an algebraic differential equation, while here (sorry, presumably) no relations between all those monomials could be possible. Maybe, this is more concrete... | |
| Apr 30, 2010 at 11:54 | comment | added | Wadim Zudilin | Martin, I was finally surprised by the prize... The set of differrence-differential monomials in a fixed weight will be finite (if the set of operators is finite); in your starting example $|\lambda|$ could be such a weight. The most natural (to me!) example, the ring of quasimodular forms $\mathbb Q[E_2,E_4,E_6]$, has a different grading. I am trying to write down explicitly those $E_j(q^2)$ by means of $E_j(q)$ (for seeing how $D_2$ acts) but it seems to be nontrivial enough. After submitting the answer I've realized that multiple zeta values might be a better example. | |
| Apr 30, 2010 at 11:04 | comment | added | Martin Rubey | Wadim, many thanks for your answer! I admit that I accepted without reading all details, to make sure the bounty is properly awarded. Thus, could you expand a bit? In particular, it is not yet clear to me which monomials I should/could consider, and why this set of monomials would be finite if I only consider (say) D and D_2. Already the set of monomials in the usual differentiation operator is infinite, so surely I misunderstood... | |
| Apr 30, 2010 at 6:49 | vote | accept | Martin Rubey | ||
| Apr 30, 2010 at 6:49 | history | bounty ended | Martin Rubey | ||
| Apr 30, 2010 at 4:03 | comment | added | Wadim Zudilin | Yes Will, I follow that question and have realized that your answer is not there any more. I would be happy to see a clear algorithm but for the moment the best answer is still too theoretical to me to be implemented (even I am not the author of the problem). | |
| Apr 30, 2010 at 1:20 | comment | added | Will Jagy | Hi Wadim, did you see the answers to mathoverflow.net/questions/22777 on the lattice point close to an intersection? | |
| Apr 30, 2010 at 0:52 | history | answered | Wadim Zudilin | CC BY-SA 2.5 |