ag.algebraic geometry – Localizations of Smooth Spherical Varieties at Simple Roots

Setup

Let $G$ be a (connected) reductive group over an algebraically closed field $k$, and fix a Borel subgroup
$B subset G$ and a maximal torus $T subset B$. Let $lambda: mathbb{G}_m to T$ be a dominant one-parameter subgroup. For any complete normal $G$-variety $X$ (by “variety” I mean separated, integral, and finite-type over $k$), it is a general fact that the Białynicki-Birula decomposition of $X$ with respect to $lambda$ has a big cell. In other words, there exists a connected component $S subset X^{mathbb{G}_m}$ such that $lim_{t to 0} lambda(t)x in S$ for all $x$ in some open subset of $X$. Moreover, $S$ is a normal variety and carries an action of the reductive group $M_lambda = {g in G | lambda(t)g = glambda(t) forall t in mathbb{G}_m}$.

I am interested in the case where $mathrm{char}(k) = 0$ and $X$ is a smooth projective spherical $G$-variety. Then, $S$ is a spherical $M_lambda$-variety, and I believe $S$ is also smooth and projective (because the fixed point scheme $X^{mathbb{G}_m}$ is closed, and Iversen has proven that $X^{mathbb{G}_m}$ is smooth). When $X$ is in fact a toroidal spherical variety, $S$ is also toroidal, and a nice paper by Knop shows that all the combinatorial data of the spherical variety $S$ (i.e. its spherical homogeneous datum and its colored fan) are completely determined by the combinatorial data of $X$.

My Question: what can be said about the combinatorial data of $S$ when $X$ is smooth and projective but not necessarily toroidal?

A Few Ideas

It seems likely to me that Knop’s results for toroidal varieties should at least partly hold in this setting. For one thing, it is well-known that there exists a $G$-equivariant birational morphism
$pi: X’ to X$ with $X’$ complete and toroidal. Knop’s results hold for $X’$, and it seems to me that some of these results should pass down to $X$. I am also thinking that the local structure theorem along with Luna’s étale slice theorem may be enough to repeat at least some of Knop’s arguments for the toroidal case. Indeed, some of Knop’s arguments use the local structure theorem on an open subset $U subset X$ to get $U cong R_u(P) times Z$, where $P = {g in G | gU = U}$ and $Z$ is an affine spherical variety under the action of a Levi subgroup $M subset P$. In the toroidal case, $Z$ is in fact toric for a quotient of $M$, which is the key to Knop’s argument. In the smooth but not necessarily toroidal case, $Z$ is smooth affine and spherical, so Luna’s étale slice theorem gives $Z cong M times^H V$, where $H$ is reductive, $M/H$ is an affine spherical $M$-variety and $V$ is a spherical $H$-module (see Corollary 2.2 of this paper). So, it seems like the structure of $Z$ might still be nice enough to relate $lim_{t to 0} lambda(t)z$ for $z in Z$ to the combinatorial data of $Z$ and $X$.

I intend to try to work out these approaches in more detail myself, but I’m not particularly familiar with these sorts of arguments, so I would really appreciate any help, intuition, or related results that anyone is willing to offer!

discrete mathematics – Find all four square roots of 61 in Z247, and show that x^3737 = x for all x ∈ Z247.

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reference request – Name for a logarithmic ratio of roots

I’m trying to find a name for the following quantity that came up in my research. I’ve asked some people and looked around myself but can’t find a name, yet it seems like something that has probably been studied before.

Let $K$ be a field and $f(x)$ a polynomial in $K(x)$, and further assume that $bar K$ has a norm $|cdot |$. Let $alpha,beta$ be the roots of smallest and largest norm, respectively. What would one call, and are there any references you know of related to:
$$frac{log|alpha|}{log|beta|}?$$
(or its reciprocal, of course)

I’m mainly interested in the case where $K$ is a local field, so really $log|alpha|$ is the valuation of $alpha$ – for this reason I prefer not to assume that $f(x)$ is irreducible over $K$.

polynomials – Roots given by Solve are not satisfied by the equation

You are running into a problem because of a loss of numerical precision in your calculations.

Let me change your code to use arbitrary precision throughout:

delta = 10;
lund = 10^4;
alpha = 10^(-7);
g = 10;
eta = 1;
VA = lund*eta/delta;
k0 = Sqrt(6)/delta;

ks = 1/Sqrt(2) k0;
kz = Sqrt(k0^2 - ks^2);
k = Sqrt(ks^2 + kz^2);

beta = 10^(-2)*VA^2*kz^2*k^2/(alpha*g*ks^2);

Omega = VA/(6/10000*delta);
omgM = VA*kz;
omge = eta*k^2;
omgA2 = (Sqrt(-g*alpha*beta)*ks/k)^2;
omgO = 2*Omega*kz/k;

eqn(l_) = 
  I l^5 + 2 l^4 omge + omgA2 omge omgM^2 - 
   2 l^2 omge (omgA2 + omgM^2 + omgO^2) - 
   I l^3 (omgA2 + omge^2 + 2 omgM^2 + omgO^2) + 
   I l (omgM^4 + omgA2 (omge^2 + omgM^2) + omge^2 omgO^2);

Let’s now Solve at arbitrary precision, then convert the numerical values of the roots using arbitrary-precision math in N, with a number of digits of precision equivalent to machine-precision numbers, but all the while keeping track of precision using the arbitrary precision machinery:

N(
  eqn(l) /. Solve(FullSimplify(eqn(l)) == 0, l), 
  $MachinePrecision
)

(* Out: {0.*10^-43 + 0.*10^-42 I, 0.*10^-61 + 0.*10^-61 I, 
         0.*10^-64 + 0.*10^-65 I, 0.*10^-61 + 0.*10^-61 I, 
         0.*10^-43 + 0.*10^-42 I}  *)

Chop(%)

(* Out: {0, 0, 0, 0, 0} *)

How to find all numerical complex roots of a non-algebraic function?

The problem arises from solving non-algebraic equation, say I have a equation $x e^x =1$, every body knows that its solution is Lambert function with an integer indices, i.e. $W_k(1)$ , where $kinmathbb{N}$. But Mathematica (Solve) gives only principal value $W_0(1)$; if one tries FindInstance, it gives $W_{-24}(1)$; meanwhile if you count the roots by CountRoots, it shows a completely wrong information, “$x e^x =1$ is not a univariate function of $x$“. Why does the program treat the index as a variable, even it is fixed?
People, who knows e.g. Bessel’s function, may understand what I am talking about, $J_{nu}(x)$ is completely different with $W_k(x)$, where $nu$ is unknown, while $k$ is fixed.

My question is more general, how to calculate numeric roots of a non-algebraic equation, in particular for multivalued function, e.g. $J_1left(e^{-1/x}right)=1$?

number theory – Find roots of a specific discontinuous trigonometric function

Given the following function $f(x)$ and an arbitrary positive non-prime integer $N$

$ { f(x) = sin(Npi/x) + cos(2Npi/x) + |sin(3Npi/x)| + |cos(5Npi/x)| + 1, N in mathbb{Z}, x in mathbb{R} } $

I am looking for any root in the range $2 <= x < N$, I am pretty sure all $f(x)$ are greater than or equal to zero, so the roots are global minima. All the roots are also points of discontinuity and their $x in mathbb{Z} $.

Question

Is there any, even numerical way, to go about finding these roots?

Here is a plot of $f(x)$ for $N=517$. It isn’t very good, its hard to choose a zoom level which will demonstrate all the features. But you can see I marked a zero at $x=22$, you will notice it looks like there is a zero at $x approx 26.5$ but it is actually $ f(x) approx 0.0011 $ if one were to zoom in.

There are many of these “close” to zero points, but none are actually zero, in the range I’m interested in ($ 2<=x<N $), unless 2 conditions I have found are met:

  1. $x in mathbb{Z} $
  2. $GCD(x,N) > 1 $

My example $N=517$ the roots are $ x in {-2,22,94} $, we don’t care about $-2$ and $GCD(22,517)=11$ and $GCD(94,517) = 47$.

I don’t think the GCD observation will help in finding the roots, but I mention it because I think its interesting.

Plot of function for N = 517

plotting – Add vertical lines to `ComplexPlot3D` at roots

I have a Table of 6 plots of polynomials of increasing degree, using ComplexPlot3D:

Clear["Global`*"]; 
poly[z_] := Sum[k*z^k, {k, 1, n}]; 
Table[ComplexPlot3D[poly[z], {z, -1.5 - 1.5*I, 1.5 + 1.5*I}], {n, 1, 6}]

enter image description here

I would like to add a black vertical line passing through each root of the polynomials. You can kind of see where they are from the plots, but lines would be a helpful visualisation aid.

I can obtain the roots easily enough:

Table[{poly[z], Roots[poly[z] == 0, z]}, {n, 1, 6}]

But how do I convert the data provided by Roots into vertical lines? Ultimately, I want to be able to do this for polynomials of arbitrary degree, so a ‘manual’ solution isn’t much help.

Thanks in advance, and stay safe.

beginner – Python code to find the roots of any integer

Hi there!
I’m new to programming and python,this one of biggest codes in python yet…
I just wanted to know if you can simplify it further…
English is not my first language, so corrections in the language are also welcomed!
Thank You In advance

x = int(input('Enter an integer who's root value you need: '))

e = int(input('Enter an +ve integer for root value: '))

#Removing the possiblities of zero in eihter of the variables

if x==0:
    print('Zero power anything is zero!')
elif e==0:
    print('Anything power zero is one!')
else:
    while e < 0:
        print('I SAID +VE INTEGER FOR ROOT!')
        x = int(input('Enter an integer who's root value you need: '))
        e = int(input('Enter an +ve integer for root value: '))

    while e %2 == 0 and x < 0:
        print('-ve nos. can't have root value for even roots!')
        x = int(input('Enter an integer who's root value you need: '))
        e = int(input('Enter an +ve integer for root value: '))
    epsilon = 0.0000001
    numGuesses = 0
    if x>0:
        low = -x
    elif x<0:
        low = x
    if x>0:
        high = max(1.0,x)
    elif x<0:
        high = max(1.0,-x)
    ans = (high + low)/2.0
    while abs(ans**e - x) >= epsilon:
        print('low =', low, 'high =', high, 'ans =', ans)
        numGuesses += 1
        if ans**e < x:
            low = ans
        else:
            high = ans
        ans = (high + low)/2.0
    print('numGuesses =', numGuesses)
    print(ans, 'is close to the value of', e, 'root', x)