In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coefficients. This motivates that, in applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots.
In the case of univariate polynomials, the product rule implies that, if p2 divides f, then p divides the formal derivative f ' of f. The converse is also true and hence,
f
{\displaystyle f}
is square-free if and only if
1
{\displaystyle 1}
is a greatest common divisor of the polynomial and its derivative.A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials
f
=
a
1
a
2
2
a
3
3
⋯
a
n
n
=
∏
k
=
1
n
a
k
k
{\displaystyle f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{n}^{n}=\prod _{k=1}^{n}a_{k}^{k}\,}
where those of the ak that are non-constant are pairwise coprime square-free polynomials (here, two polynomials are said coprime is their greatest common divisor is a constant; in other words that is the coprimality over the field of fractions of the coefficients that is considered). Every non-zero polynomial admits a square-free factorization, which is unique up to the multiplication and division of the factors by non-zero constants. The square-free factorization is much easier to compute than the complete factorization into irreducible factors, and is thus often preferred when the complete factorization is not really needed, as for the partial fraction decomposition and the symbolic integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra.
Over a field of characteristic 0, the quotient of
f
{\displaystyle f}
by its GCD with its derivative is the product of the
a
i
{\displaystyle a_{i}}
in the above square-free decomposition. Over a perfect field of non-zero characteristic p, this quotient is the product of the
a
i
{\displaystyle a_{i}}
such that i is not a multiple of p. Further GCD computations and exact divisions allow computing the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a better algorithm is known, Yun's algorithm, which is described below. Its computational complexity is, at most, twice that of the GCD computation of the input polynomial and its derivative. More precisely, if
T
n
{\displaystyle T_{n}}
is the time needed to compute the GCD of two polynomials of degree
n
{\displaystyle n}
and the quotient of these polynomial by the GCD, then
2
T
n
{\displaystyle 2T_{n}}
is an upper bound for the time needed to compute the square free decomposition.
There are also known algorithms for the computation of the square-free decomposition of multivariate polynomials, that proceed generally by considering a multivariate polynomial as a univariate polynomial with polynomial coefficients, and applying recursively a univariate algorithm.
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Requirements: Android 10+ And Shizuku app ( Important!)
Setting Up Shizuku On Android 11
Setting Up Shizuku On Android 10
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Developer
Released on
Updated
Size
Version
Version
N/A
Jan 14, 2022
Jan 14, 2022
20MB
(no root)
Varies with device
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YøùTùbé ReVanced
│ Android: 8.0 and up #update
Extended + RVP V2.199.1
ρrémíùm Features Of YøùTùbé
Completely free no ads
➠ How to Install:
★ First install MicroG
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★ Open YøùTùbé ReVanced
★ Log in to your Google account
| Release by ModzNoLimit
Download link...
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Android 11+ NON Root Only
Instruction
_____________
1. Install HttpCanary 2.7.0 A11 apk
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Certificate Install Instruction
________________________________...
Hello everyone, We've all experienced the frustration of having our phone locked and not knowing the pattern, password, or PIN. Don't worry, there's a way to unlock your device without having to perform a factory reset. This method should work on all devices with any Android version, except for...
Dead Cell 3.0.11 non root mod (100%)
Dead Cells 3.0.11 (4 episode) Mod Non root
Dead Cells Mod 3.0.11 is a famous mobile game. It is a good combination of role-playing, action & adventure genres. Do you already know about this roguevania action platformer game? It requires players to explore...
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AdAway - No Root Ad Blocker v6.1.0 [Final] [Mod] [Full Multi]
#10330703 by derrin
AdAway - No Root Ad Blocker v6.1.0-20230326 [Final] [Mod] [Full Multi]
Requirements: 8.0 and up | root optional
Overview: AdAway is an open source ad blocker for Android.
A Free and Open Source ad blocker...
Wavelet: headphone specific EQ will be an application that can give its users a great audio experience. If you are an extremely passionate music person with great music with melodious melodies, this is definitely an application that can help you a lot. Coming to this application, users will be...
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PC
Working ADB/Fastboot
Package name viewer 2.0 app (download in playstore)
any text editor app (recommended Quickedit from playstore)
1. Install package name viewer
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Spotify ρrémíùm Apk features:
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Mga kailangan na App. Nasa Playstore lahat:
1. Termux
2. AnLinux: Run Linux on Android Without Root Access
3. VNC Viewer: Remote Desktop
4. häçker's Keyboard
STEP #1
Punta sa Termux at i-type ang:
apt-get update [enter]
.....wait then type:
apt-get upgrade [enter]
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Y...
Hello po mga sir/maam
Ang tutorial na ito ay kung paano maishare ang internet ni injector para sa ibang phone/tablet/pc no need to root
1st step: idownload ang >>PdaNet+<< link: ttp://pdanet.co/a/ meron po siya sa Play store.
Iinstall ang PdaNet+ sa device na gustong mag...
Auto clicker [no root]
https://play.google.com/store/apps/details?id=simplehat.clicker
May instructions na sa app kung pano gamitin
For android 7.0 and up
Wifi WPS Unlocker
This application recover the default WPS PIN.The application was developed for educational purposes.In this application you can check if your router is vulnerable or not.if you realize someone has security problem you must inform immediately.I am not responsible for the misuse...
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Requirements:
PC or LAPTOP
MirrorGo App -...
Brevent.
In short, Prevent without Root. You can get it from here: https://play.google.com/store/apps/details?id=me.piebridge.brevent or http://piebridge.me/br/, and source code: https://github.com/brevent/Brevent
There are several ways to prevent running for apps:
- Force-Stop, will set...
AdClear is a non-root ad blocker for Android devices. There’s only one version of the app, and it’s free. We released it at the end of December and are looking for feedback and help in making AdClear the best it can be.
Features
Blocks encrypted ads
Works with Wi-Fi and cell data
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Requires Android 5.0 and up
Featured on Google Play Home Page, Android Police, Yahoo News, CNET, Android Central, Droid-Life and more.
AZ Screen Recorder is the best app to record the screen of your Lollipop. It does NOT require root access, no time limit, no watermark, ad free and...
Kailangan niyo lahat gawin yung procedure kung hindi niyo
Susundin yung mga procedure hindi siya gagana kailangan lamg mag basa ng mabuti !
Now lets start !
PROCEDURE:
1.Go to setting hanapin niyo ang "DEVELOPER OPTION" kung wala pa kayo dev. Option Go to setting "ABOUT PHONE" then click...