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Sep
19
comment How many simultaneous polynomial equations of degree 2 can software solve today?
I'm certainly not an expert here, but I'll note the following from mathworks.co.uk/help/symbolic/mupad_ref/… : "numeric::polysysroots is a hybrid routine: it calls the symbolic solver solve(eqs, vars, BackSubstitution = FALSE) and processes its symbolic result numerically." In other words, this is trying to solve the system of equations algebraically, I think, and then computing the numerical answer at the end. This technique is surely hopeless at this size, but I'd guess that genuinely numerical techniques should do fairly well.
Sep
19
comment Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant
Many thanks for this Yemon - this answers the "do they have a name" part of my question. It turns out arxiv.org/pdf/1311.5390v2.pdf includes what is essentially the proof of Sean's result below: in fact, they both reference the same MO post. I'd be very interested if anyone familiar with that literature could clarify the state of the art, maybe for prime $N$.
Sep
19
revised Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant
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Sep
17
awarded  Nice Question
Sep
17
asked Functions $f$ on $\mathbb{Z}/N\mathbb{Z}$ with $|f|$ and $|\widehat{f}|$ constant
Apr
13
comment Is there a “right” proof of Riemann's Theta Relation?
@J. Martel: I have to confess I started to get a bit out of my algebraic depth with the later (adelic) stuff in Tata III, so haven't yet managed to get a working understanding of Thm 7.4, but will persevere. Possibly I should just try to figure out what that result is saying in the Euclidean case. @Jeff: That would be great! @Everyone: the equation for $x_1$ in Mumford looks like it must be a typo; corrected in the above.
Apr
13
revised Is there a “right” proof of Riemann's Theta Relation?
added 12 characters in body
Apr
12
asked Is there a “right” proof of Riemann's Theta Relation?
Apr
9
comment Minimal size of subsets $A,B$ in a finite group $G$ such that $AB=G$
So, the strategy when A = H is a subgroup generalizes to A being your favorite set with $|A A^{-1}| = O(|A|)$, by a similar packing argument (if you accept a constant factor at the end). The aim is then to find such a set with $|A| \approx \sqrt{N}$. This gives rise to the cyclic example above, and should do what Colin Reid wants for nilpotent groups (see above). I don't know Whether you could use something like this to avoid the use of CFSG in the general case.
Mar
22
awarded  Citizen Patrol
Mar
22
awarded  Critic
Mar
22
awarded  Commentator
Mar
22
comment More expanders?
My guess is that (1) and (2) are doomed not to expand because they are "too abelian", or "too nilpotent". So for (1), a set like $\{\pm g^{n} : 0 \le n \le N \}$ for $N$ large (but small compared to $p$) is likely to contradict expansion. Something similar should work in (2), considering a set of bounded length words on both your generators. I think this is closely related to this MO question. (3) has a very different flavour. It might be related to expansion in $SL_2$.
Mar
21
comment Joint (close to uniform) distribution in finite fields
I think that's right -- third edit lucky. I think the $L^2$ result should give $\varepsilon$ up to a (basically) constant factor like $k / (k - 1)$, meaning the $L^1$ result should go as $k \varepsilon$ times some constant factor; i.e. this hasn't made too much difference to the conclusion.
Mar
21
revised Joint (close to uniform) distribution in finite fields
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Mar
21
revised Joint (close to uniform) distribution in finite fields
added 192 characters in body
Mar
21
comment Joint (close to uniform) distribution in finite fields
Kevin -- thanks! You're absolutely right, and that means there's also an $\varepsilon^2$ that needs tracing through the argument. I'll fix this now and see what I get.
Mar
21
revised Joint (close to uniform) distribution in finite fields
typo
Mar
21
answered Joint (close to uniform) distribution in finite fields
Mar
20
awarded  Yearling