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\title{Memorial University\\Grenfell Campus\\ Math 4950\\Quantifier Elimination following Muchnik Sets}
\author{Purvashik Raghoonundun}
\date{20/04/2019}
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\tableofcontents
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\begin{abstract}
This paper is a detailed work of Christian Michaux and Adem Ozturk on Quantifier Elimination following Muchnik Sets.The paper describes a simple algorithm of quantifier elimination for real closed fields and algebraically closed fields following an idea of Muchnik. It comprises of determining from the coefficients of a polynomial, a finite list of polynimial expression in theses coefficients, that is the understanding of the signs of these expressions will help us to understand and know the sign table of the original polynomial.Some simple procedures like Eucleadean division,differientiation, evaluation, ommiting of the leading terms of polynomials and intermediate value property are used in this paper. Furthermore, the algorithm can be modified and turned into an algorithm of quantifier elimination for algebraically closed fields too.
\end{abstract}
\section{Introduction}
\subsection{Muchnik's Algorithm}
In this paper, we will consider polynomials in the form of $p(\overrightarrow{Y},X)$ with coefficients in $\mathbb{Z}$ and where $X$ and $\overrightarrow{Y} = Y_{1},Y_{2},...,Y_{m}$ are variables. Such polynomials can also be considered as polynomials in the variable $X$ with coefficients in the ring $\mathbb{Z}[Y_{1},Y_{2},...,Y_{m}]$. The notation deg $p$ stands for the degree of $p(\overrightarrow{Y},X)$ with respect to the variable $X$. In this algorithm, we will use the following four operations which apply to polynomials with non zero degree.\\
Let $p(X) = a_{n}X^{n} + a_{n-1}X^{n-1} +... + a_{0} \in$ $\mathbb{Z}[\overrightarrow{Y}][X]$ with $n \ge 1$ and $a_{n} \neq 0$.\\
i$)$ Derivative: $D(p) = \frac{\partial}{\partial X}p$ \\
Example: Let $p= 3x^{2} + x +1 $
$$\frac{\partial p}{\partial x} = 6x + 1$$ \\
ii$)$ Extracting the leading coefficient: $E(p) = a_{n}$.\\ So, for $p= 3x^{2} + x +1 $; $$E(p) = 3$$\\
iii$)$ Omission of the leading term: $Om(p) = a_{n-1}X^{n-1} +... + a_{0}$ where $a_{n}$ is omitted.\\
When $p= 3x^{2} + x +1 $; $$Om(p) = x +1$$ \\
iv$)$ Modified remainder: Modified remainders, also known as pseudo remainders.We perform Euclidean division(i.e division of 2 integers which gives a quotient and a remainder smaller than the division), however, in this problem, it will be the division of two polynomials but with some special rules.Let $p(X) = a_{n}X^{n} +... + a_{0}$ and $q= b_{m}X^{m} + ...+ b_{0}$ such that $n$ is the degree of polynomial $p$, denoted by deg $p$ which is greater or equal to $m$ which is the degree of polynomial $q$, denoted by deg $q$ and $p \neq q$.Then, there are unique $r$ and $h \in \mathbb{Z}[Y_{1},Y_{2},...,Y_{m}]$ with $r$ of degree less than deg $q$ such that:$$b_{m}^{n-m+1}p = q\cdot h + r,$$ where b{m} is the leading coefficient of polynomial $q$ , $d$ is the degree of $p$ and $m$ is the degree of $q$.\\
We represent $r$, the pseudo remainder/modified remainder of $p$ and $q$ as $RM(p,q)$.
The pseudo remainder,r can be calculated by using the long division method. By taking the long division of $b_{m}^{n-m+1}p$ and $q$, we get the value $h$ and $r$.
Example 1:
Let $p= 3x^{2} + x +1 $
and take the derivative of $p$ which $6x+1$ as $q$
The first step is to check the degrees. If the degree of p is less or equal than $q$ and $p=q$, then we can perform the division to find the pseudo remainder.\\ Denote degree of p as deg $p$ and denote the degree of q as deg $q$.\\deg$p = 2$ and deg $q = 3$. Since deg $p >$ deg$q$ then we move to the second step.
The second step is to use the equation $b_{m}^{n-m+1}p = q\cdot h + r.$ where $b_{m} = 6 , n=2$ and $m=1.$ We replace the values in the equation and get
\begin{equation}
\begin{split}
6^{2-1+1}\cdot (3x^{2} + x +1 )& = h \cdot (6x+1) + r \\
6^{2}\cdot (3x^{2} + x +1 )& = h \cdot (6x+1) + r\\
36 \cdot (3x^{2} + x +1 )& = h \cdot (6x+1) + r\\
108x^{2} + 36x +36 &= h \cdot (6x+1) + r\\
\end{split}
\end{equation}
Then we use the long division between $(108x^{2} + 36x +36)$ and $(6x+1)$ to get the value of $h$ and $r$.
\longdiv{108x^2+36x+36}{6x+1}
\section{Conclusion}
Write your conclusion here.
\end{document}
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