the degree of the multiplier of Misiurewicz points

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Define the iterated complex quadratic polynomial
$$beginalignedf^0phantom+1_c(z) &= z \ f_c^n + 1(z) &= (f_c^n(z))^2+cendaligned$$



A Misiurewicz point $M_q,p$ satisfies
$$f_c^q + p(0) = f_c^q(0)$$
where $q > 0$ is the preperiod and $p > 0$ is the period. The equation also has roots with lower preperiod (including $0$) and/or period, these should be discounted. By construction each $M_q,p$ is an algebraic integer. The multiplier $m$ of $c = M_q,p$ is defined as
$$ m = prod_n = q^q + p - 1 2 f_c^n(0) $$
and is also an algebraic integer.



Question: is the degree of the minimal polynomial of $m$ always equal to the degree of the minimal polynomial of $M_q,p$?



Here is a table of degrees of minimal polynomials of some $M_q,p$:



q p 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 1 3 6 15 27 63 120 252 495 1023 2010 4095
1 0 0 0 0 0 0 0 0 0 0 0 0
2 1 2 6 12 30 54 126 240 504 990 2046
3 3 3 12 24 60 108 252 480 1008 1980
4 7 8 21 48 120 216 504 960 2016
5 15 15 48 90 240 432 1008 1920
6 31 32 96 192 465 864 2016
7 63 63 189 384 960 1701
8 127 128 384 768 1920
9 255 255 768 1530
10 511 512 1533
11 1023 1023
12 2047


Calculated with this Haskell code:



-
Prints tables about Misiurewicz points in the Mandelbrot set.
Degree of the polynomials
-

-# LANGUAGE NoImplicitPrelude #-
-# LANGUAGE FlexibleContexts #-

import NumericPrelude hiding (divMod)
import MathObj.Polynomial
import MathObj.Polynomial.Core hiding (divMod, divModRev)
import Data.Tuple.HT (mapPair, mapFst, forcePair)
import Data.List.HT (switchL)
import qualified NumericPrelude.Base as P
import qualified Data.List as List

import Data.MemoTrie (memo, memo2)

import System.Environment (getArgs)
import System.IO (hPutStrLn, stderr)

table :: [[String]] -> String
table = unlines . map (List.intercalate "t")

td :: Int -> [[String]]
td n = (( "deg M_q,p" : map show [ 1 .. n ]) :
 [ map show $ q : [ d q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

tp :: Int -> [[String]]
tp n = (( "M_q,p": map show [ 1 .. n ]) :
 [ show q : [ show . coeffs $ m q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

main' :: Int -> Int -> IO ()
main' n1 n2 = do
 putStr . table . td $ n1
 putStr . table . tp $ n2


main :: IO ()
main = do
 args <- map reads `fmap` getArgs
 case args of
 [[(n1,"")], [(n2,"")]] -> main' n1 n2
 _ -> hPutStrLn stderr "expected two integer arguments (eg 8 4)"

type P = T Integer

divideAll :: P -> P -> P
divideAll h g
 | isZero h = h
 | isOne g = h
 | isZero g = error "/0"
 | otherwise = case h `divMod` g of
 (di, mo)
 | isZero mo -> di `divideAll` g
 | otherwise -> h

divideAlls :: P -> [P] -> P
divideAlls h = h
divideAlls h (g:gs) = divideAlls (h `divideAll` g) gs

c :: P
c = fromCoeffs [ 0, 1 ]

f :: P -> P
f z = z^2 + c

fn :: Int -> P
fn = memo fn_
 where
 fn_ 0 = 0
 fn_ n = f (fn (n - 1))

m_raw :: Int -> Int -> P
m_raw = memo2 m_raw_
 where
 m_raw_ q p = fn (q + p) - fn q

m :: Int -> Int -> P
m = memo2 m_
 where
 m_ q p = fromCoeffs . normalize . coeffs $
 m_raw q p `divideAlls`
 [ mqp
 | q' <- [ 0 .. q ]
 , p' <- [ 1 .. p ]
 , q' + p' < q + p
 , p `mod` p' == 0
 , let mqp = m q' p'
 , not (isZero mqp)
 ]

d :: Int -> Int -> Int
d q p = case degree (m q p) of Just k -> k ; Nothing -> -1

isOne x = isZero (x - one)

-
the following is copy pasted from the source of
<https://hackage.haskell.org/package/numeric-prelude-0.4.3/docs/MathObj-Polynomial-Core.html>
with one minor modification: to assert y0=1 and omit the division /y0
this allows it to work with monic polynomials with Integer coefficients
-

divMod x y = mapPair (fromCoeffs, fromCoeffs) $ divMod1 (coeffs x) (coeffs y)

--divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divMod1 x y =
 mapPair (List.reverse, List.reverse) $
 divModRev1 (List.reverse x) (List.reverse y)

-
snd $ Poly.divMod (repeat (1::Double)) [1,1]
-
- 
--divModRev :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divModRev1 x y =
 case dropWhile isZero y of
 -> error "MathObj.Polynomial: division by zero"
 y0:ys | isOne y0 ->
 let -- the second parameter represents lazily (length x - length (normalize y))
 aux xs' =
 forcePair .
 switchL
 (, xs')
 (P.const $
 let (x0:xs) = xs'
 q0 = x0
 in mapFst (q0:) . aux (sub xs (scale q0 ys)))
 in aux x (drop (length ys) x)
 _ -> error "MathObj.Polynomial: division by non-monic"



algebraic-number-theory




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  • math.stackexchange.com/questions/2740655/… related
    – Claude
    17 hours ago














up vote
1
down vote

favorite
1












Define the iterated complex quadratic polynomial
$$beginalignedf^0phantom+1_c(z) &= z \ f_c^n + 1(z) &= (f_c^n(z))^2+cendaligned$$



A Misiurewicz point $M_q,p$ satisfies
$$f_c^q + p(0) = f_c^q(0)$$
where $q > 0$ is the preperiod and $p > 0$ is the period. The equation also has roots with lower preperiod (including $0$) and/or period, these should be discounted. By construction each $M_q,p$ is an algebraic integer. The multiplier $m$ of $c = M_q,p$ is defined as
$$ m = prod_n = q^q + p - 1 2 f_c^n(0) $$
and is also an algebraic integer.



Question: is the degree of the minimal polynomial of $m$ always equal to the degree of the minimal polynomial of $M_q,p$?



Here is a table of degrees of minimal polynomials of some $M_q,p$:



q p 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 1 3 6 15 27 63 120 252 495 1023 2010 4095
1 0 0 0 0 0 0 0 0 0 0 0 0
2 1 2 6 12 30 54 126 240 504 990 2046
3 3 3 12 24 60 108 252 480 1008 1980
4 7 8 21 48 120 216 504 960 2016
5 15 15 48 90 240 432 1008 1920
6 31 32 96 192 465 864 2016
7 63 63 189 384 960 1701
8 127 128 384 768 1920
9 255 255 768 1530
10 511 512 1533
11 1023 1023
12 2047


Calculated with this Haskell code:



-
Prints tables about Misiurewicz points in the Mandelbrot set.
Degree of the polynomials
-

-# LANGUAGE NoImplicitPrelude #-
-# LANGUAGE FlexibleContexts #-

import NumericPrelude hiding (divMod)
import MathObj.Polynomial
import MathObj.Polynomial.Core hiding (divMod, divModRev)
import Data.Tuple.HT (mapPair, mapFst, forcePair)
import Data.List.HT (switchL)
import qualified NumericPrelude.Base as P
import qualified Data.List as List

import Data.MemoTrie (memo, memo2)

import System.Environment (getArgs)
import System.IO (hPutStrLn, stderr)

table :: [[String]] -> String
table = unlines . map (List.intercalate "t")

td :: Int -> [[String]]
td n = (( "deg M_q,p" : map show [ 1 .. n ]) :
 [ map show $ q : [ d q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

tp :: Int -> [[String]]
tp n = (( "M_q,p": map show [ 1 .. n ]) :
 [ show q : [ show . coeffs $ m q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

main' :: Int -> Int -> IO ()
main' n1 n2 = do
 putStr . table . td $ n1
 putStr . table . tp $ n2


main :: IO ()
main = do
 args <- map reads `fmap` getArgs
 case args of
 [[(n1,"")], [(n2,"")]] -> main' n1 n2
 _ -> hPutStrLn stderr "expected two integer arguments (eg 8 4)"

type P = T Integer

divideAll :: P -> P -> P
divideAll h g
 | isZero h = h
 | isOne g = h
 | isZero g = error "/0"
 | otherwise = case h `divMod` g of
 (di, mo)
 | isZero mo -> di `divideAll` g
 | otherwise -> h

divideAlls :: P -> [P] -> P
divideAlls h = h
divideAlls h (g:gs) = divideAlls (h `divideAll` g) gs

c :: P
c = fromCoeffs [ 0, 1 ]

f :: P -> P
f z = z^2 + c

fn :: Int -> P
fn = memo fn_
 where
 fn_ 0 = 0
 fn_ n = f (fn (n - 1))

m_raw :: Int -> Int -> P
m_raw = memo2 m_raw_
 where
 m_raw_ q p = fn (q + p) - fn q

m :: Int -> Int -> P
m = memo2 m_
 where
 m_ q p = fromCoeffs . normalize . coeffs $
 m_raw q p `divideAlls`
 [ mqp
 | q' <- [ 0 .. q ]
 , p' <- [ 1 .. p ]
 , q' + p' < q + p
 , p `mod` p' == 0
 , let mqp = m q' p'
 , not (isZero mqp)
 ]

d :: Int -> Int -> Int
d q p = case degree (m q p) of Just k -> k ; Nothing -> -1

isOne x = isZero (x - one)

-
the following is copy pasted from the source of
<https://hackage.haskell.org/package/numeric-prelude-0.4.3/docs/MathObj-Polynomial-Core.html>
with one minor modification: to assert y0=1 and omit the division /y0
this allows it to work with monic polynomials with Integer coefficients
-

divMod x y = mapPair (fromCoeffs, fromCoeffs) $ divMod1 (coeffs x) (coeffs y)

--divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divMod1 x y =
 mapPair (List.reverse, List.reverse) $
 divModRev1 (List.reverse x) (List.reverse y)

-
snd $ Poly.divMod (repeat (1::Double)) [1,1]
-
- 
--divModRev :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divModRev1 x y =
 case dropWhile isZero y of
 -> error "MathObj.Polynomial: division by zero"
 y0:ys | isOne y0 ->
 let -- the second parameter represents lazily (length x - length (normalize y))
 aux xs' =
 forcePair .
 switchL
 (, xs')
 (P.const $
 let (x0:xs) = xs'
 q0 = x0
 in mapFst (q0:) . aux (sub xs (scale q0 ys)))
 in aux x (drop (length ys) x)
 _ -> error "MathObj.Polynomial: division by non-monic"



algebraic-number-theory




share|cite|improve this question





















  • math.stackexchange.com/questions/2740655/… related
    – Claude
    17 hours ago












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





Define the iterated complex quadratic polynomial
$$beginalignedf^0phantom+1_c(z) &= z \ f_c^n + 1(z) &= (f_c^n(z))^2+cendaligned$$



A Misiurewicz point $M_q,p$ satisfies
$$f_c^q + p(0) = f_c^q(0)$$
where $q > 0$ is the preperiod and $p > 0$ is the period. The equation also has roots with lower preperiod (including $0$) and/or period, these should be discounted. By construction each $M_q,p$ is an algebraic integer. The multiplier $m$ of $c = M_q,p$ is defined as
$$ m = prod_n = q^q + p - 1 2 f_c^n(0) $$
and is also an algebraic integer.



Question: is the degree of the minimal polynomial of $m$ always equal to the degree of the minimal polynomial of $M_q,p$?



Here is a table of degrees of minimal polynomials of some $M_q,p$:



q p 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 1 3 6 15 27 63 120 252 495 1023 2010 4095
1 0 0 0 0 0 0 0 0 0 0 0 0
2 1 2 6 12 30 54 126 240 504 990 2046
3 3 3 12 24 60 108 252 480 1008 1980
4 7 8 21 48 120 216 504 960 2016
5 15 15 48 90 240 432 1008 1920
6 31 32 96 192 465 864 2016
7 63 63 189 384 960 1701
8 127 128 384 768 1920
9 255 255 768 1530
10 511 512 1533
11 1023 1023
12 2047


Calculated with this Haskell code:



-
Prints tables about Misiurewicz points in the Mandelbrot set.
Degree of the polynomials
-

-# LANGUAGE NoImplicitPrelude #-
-# LANGUAGE FlexibleContexts #-

import NumericPrelude hiding (divMod)
import MathObj.Polynomial
import MathObj.Polynomial.Core hiding (divMod, divModRev)
import Data.Tuple.HT (mapPair, mapFst, forcePair)
import Data.List.HT (switchL)
import qualified NumericPrelude.Base as P
import qualified Data.List as List

import Data.MemoTrie (memo, memo2)

import System.Environment (getArgs)
import System.IO (hPutStrLn, stderr)

table :: [[String]] -> String
table = unlines . map (List.intercalate "t")

td :: Int -> [[String]]
td n = (( "deg M_q,p" : map show [ 1 .. n ]) :
 [ map show $ q : [ d q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

tp :: Int -> [[String]]
tp n = (( "M_q,p": map show [ 1 .. n ]) :
 [ show q : [ show . coeffs $ m q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

main' :: Int -> Int -> IO ()
main' n1 n2 = do
 putStr . table . td $ n1
 putStr . table . tp $ n2


main :: IO ()
main = do
 args <- map reads `fmap` getArgs
 case args of
 [[(n1,"")], [(n2,"")]] -> main' n1 n2
 _ -> hPutStrLn stderr "expected two integer arguments (eg 8 4)"

type P = T Integer

divideAll :: P -> P -> P
divideAll h g
 | isZero h = h
 | isOne g = h
 | isZero g = error "/0"
 | otherwise = case h `divMod` g of
 (di, mo)
 | isZero mo -> di `divideAll` g
 | otherwise -> h

divideAlls :: P -> [P] -> P
divideAlls h = h
divideAlls h (g:gs) = divideAlls (h `divideAll` g) gs

c :: P
c = fromCoeffs [ 0, 1 ]

f :: P -> P
f z = z^2 + c

fn :: Int -> P
fn = memo fn_
 where
 fn_ 0 = 0
 fn_ n = f (fn (n - 1))

m_raw :: Int -> Int -> P
m_raw = memo2 m_raw_
 where
 m_raw_ q p = fn (q + p) - fn q

m :: Int -> Int -> P
m = memo2 m_
 where
 m_ q p = fromCoeffs . normalize . coeffs $
 m_raw q p `divideAlls`
 [ mqp
 | q' <- [ 0 .. q ]
 , p' <- [ 1 .. p ]
 , q' + p' < q + p
 , p `mod` p' == 0
 , let mqp = m q' p'
 , not (isZero mqp)
 ]

d :: Int -> Int -> Int
d q p = case degree (m q p) of Just k -> k ; Nothing -> -1

isOne x = isZero (x - one)

-
the following is copy pasted from the source of
<https://hackage.haskell.org/package/numeric-prelude-0.4.3/docs/MathObj-Polynomial-Core.html>
with one minor modification: to assert y0=1 and omit the division /y0
this allows it to work with monic polynomials with Integer coefficients
-

divMod x y = mapPair (fromCoeffs, fromCoeffs) $ divMod1 (coeffs x) (coeffs y)

--divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divMod1 x y =
 mapPair (List.reverse, List.reverse) $
 divModRev1 (List.reverse x) (List.reverse y)

-
snd $ Poly.divMod (repeat (1::Double)) [1,1]
-
- 
--divModRev :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divModRev1 x y =
 case dropWhile isZero y of
 -> error "MathObj.Polynomial: division by zero"
 y0:ys | isOne y0 ->
 let -- the second parameter represents lazily (length x - length (normalize y))
 aux xs' =
 forcePair .
 switchL
 (, xs')
 (P.const $
 let (x0:xs) = xs'
 q0 = x0
 in mapFst (q0:) . aux (sub xs (scale q0 ys)))
 in aux x (drop (length ys) x)
 _ -> error "MathObj.Polynomial: division by non-monic"



algebraic-number-theory




share|cite|improve this question













Define the iterated complex quadratic polynomial
$$beginalignedf^0phantom+1_c(z) &= z \ f_c^n + 1(z) &= (f_c^n(z))^2+cendaligned$$



A Misiurewicz point $M_q,p$ satisfies
$$f_c^q + p(0) = f_c^q(0)$$
where $q > 0$ is the preperiod and $p > 0$ is the period. The equation also has roots with lower preperiod (including $0$) and/or period, these should be discounted. By construction each $M_q,p$ is an algebraic integer. The multiplier $m$ of $c = M_q,p$ is defined as
$$ m = prod_n = q^q + p - 1 2 f_c^n(0) $$
and is also an algebraic integer.



Question: is the degree of the minimal polynomial of $m$ always equal to the degree of the minimal polynomial of $M_q,p$?



Here is a table of degrees of minimal polynomials of some $M_q,p$:



q p 1 2 3 4 5 6 7 8 9 10 11 12 13
0 1 1 3 6 15 27 63 120 252 495 1023 2010 4095
1 0 0 0 0 0 0 0 0 0 0 0 0
2 1 2 6 12 30 54 126 240 504 990 2046
3 3 3 12 24 60 108 252 480 1008 1980
4 7 8 21 48 120 216 504 960 2016
5 15 15 48 90 240 432 1008 1920
6 31 32 96 192 465 864 2016
7 63 63 189 384 960 1701
8 127 128 384 768 1920
9 255 255 768 1530
10 511 512 1533
11 1023 1023
12 2047


Calculated with this Haskell code:



-
Prints tables about Misiurewicz points in the Mandelbrot set.
Degree of the polynomials
-

-# LANGUAGE NoImplicitPrelude #-
-# LANGUAGE FlexibleContexts #-

import NumericPrelude hiding (divMod)
import MathObj.Polynomial
import MathObj.Polynomial.Core hiding (divMod, divModRev)
import Data.Tuple.HT (mapPair, mapFst, forcePair)
import Data.List.HT (switchL)
import qualified NumericPrelude.Base as P
import qualified Data.List as List

import Data.MemoTrie (memo, memo2)

import System.Environment (getArgs)
import System.IO (hPutStrLn, stderr)

table :: [[String]] -> String
table = unlines . map (List.intercalate "t")

td :: Int -> [[String]]
td n = (( "deg M_q,p" : map show [ 1 .. n ]) :
 [ map show $ q : [ d q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

tp :: Int -> [[String]]
tp n = (( "M_q,p": map show [ 1 .. n ]) :
 [ show q : [ show . coeffs $ m q p | p <- [1 .. n], q + p <= n ]
 | q <- [0 .. n] ])

main' :: Int -> Int -> IO ()
main' n1 n2 = do
 putStr . table . td $ n1
 putStr . table . tp $ n2


main :: IO ()
main = do
 args <- map reads `fmap` getArgs
 case args of
 [[(n1,"")], [(n2,"")]] -> main' n1 n2
 _ -> hPutStrLn stderr "expected two integer arguments (eg 8 4)"

type P = T Integer

divideAll :: P -> P -> P
divideAll h g
 | isZero h = h
 | isOne g = h
 | isZero g = error "/0"
 | otherwise = case h `divMod` g of
 (di, mo)
 | isZero mo -> di `divideAll` g
 | otherwise -> h

divideAlls :: P -> [P] -> P
divideAlls h = h
divideAlls h (g:gs) = divideAlls (h `divideAll` g) gs

c :: P
c = fromCoeffs [ 0, 1 ]

f :: P -> P
f z = z^2 + c

fn :: Int -> P
fn = memo fn_
 where
 fn_ 0 = 0
 fn_ n = f (fn (n - 1))

m_raw :: Int -> Int -> P
m_raw = memo2 m_raw_
 where
 m_raw_ q p = fn (q + p) - fn q

m :: Int -> Int -> P
m = memo2 m_
 where
 m_ q p = fromCoeffs . normalize . coeffs $
 m_raw q p `divideAlls`
 [ mqp
 | q' <- [ 0 .. q ]
 , p' <- [ 1 .. p ]
 , q' + p' < q + p
 , p `mod` p' == 0
 , let mqp = m q' p'
 , not (isZero mqp)
 ]

d :: Int -> Int -> Int
d q p = case degree (m q p) of Just k -> k ; Nothing -> -1

isOne x = isZero (x - one)

-
the following is copy pasted from the source of
<https://hackage.haskell.org/package/numeric-prelude-0.4.3/docs/MathObj-Polynomial-Core.html>
with one minor modification: to assert y0=1 and omit the division /y0
this allows it to work with monic polynomials with Integer coefficients
-

divMod x y = mapPair (fromCoeffs, fromCoeffs) $ divMod1 (coeffs x) (coeffs y)

--divMod :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divMod1 x y =
 mapPair (List.reverse, List.reverse) $
 divModRev1 (List.reverse x) (List.reverse y)

-
snd $ Poly.divMod (repeat (1::Double)) [1,1]
-
- 
--divModRev :: (ZeroTestable.C a, Field.C a) => [a] -> [a] -> ([a], [a])
divModRev1 x y =
 case dropWhile isZero y of
 -> error "MathObj.Polynomial: division by zero"
 y0:ys | isOne y0 ->
 let -- the second parameter represents lazily (length x - length (normalize y))
 aux xs' =
 forcePair .
 switchL
 (, xs')
 (P.const $
 let (x0:xs) = xs'
 q0 = x0
 in mapFst (q0:) . aux (sub xs (scale q0 ys)))
 in aux x (drop (length ys) x)
 _ -> error "MathObj.Polynomial: division by non-monic"




algebraic-number-theory





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 10 hours ago
























asked 19 hours ago









Claude

2,416419




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  • math.stackexchange.com/questions/2740655/… related
    – Claude
    17 hours ago
















  • math.stackexchange.com/questions/2740655/… related
    – Claude
    17 hours ago















math.stackexchange.com/questions/2740655/… related
– Claude
17 hours ago




math.stackexchange.com/questions/2740655/… related
– Claude
17 hours ago















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What is the equation of a 3D cone with generalised tilt?

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Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
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Color the edges and diagonals of a regular polygon

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Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
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Relationship between determinant of matrix and determinant of adjoint?

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Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
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