Stamp of Sylvester

In linear algebra, the matrix of Sylvester of two Polynôme S brings information of an arithmetic nature on these polynomials. It holds its name of James Joseph Sylvester. It is used for the definition of the Résultant from two polynomials.

Definition

Are p and Q two polynomials nonnull, respective degrees m and N .

$p\; (Z)\; =p\_0+p\_1\; z+p\_2\; z^2+\; \backslash \; cdots+p\_m\; z^m,\; \backslash ;\; Q\; (Z)\; =q\_0+q\_1\; z+q\_2\; z^2+\; \backslash \; cdots+q\_n\; z^n.$

The matrix of Sylvester associated with p and Q is the square matrix $(n+m)\; \backslash \; times\; (n+m)$ thus definite

• the first line is formed of the coefficients of p , followed by 0

• the second line is obtained starting from the first by circular shift towards the line
• them (N2) following lines are obtained by repeating even operation
• the line (n+1) is formed of the coefficients of Q , followed by 0

• the following lines are formed by circular shifts

Thus in the case m =4 and N =3, the matrix obtained is

$S\_\; \{p,\; Q\}\; =\; \backslash \; begin\; \{pmatrix\}$

p_4 & p_3 & p_2 & p_1 & p_0 & 0 & 0 \ \ 0 & p_4 & p_3 & p_2 & p_1 & p_0 & 0 \ \ 0 & 0 & p_4 & p_3 & p_2 & p_1 & p_0 \ \ q_3 & q_2 & q_1 & q_0 & 0 & 0 & 0 \ \ 0 & q_3 & q_2 & q_1 & q_0 & 0 & 0 \ \ 0 & 0 & q_3 & q_2 & q_1 & q_0 & 0 \ \ 0 & 0 & 0 & q_3 & q_2 & q_1 & q_0 \ \ \end{pmatrix}.

The determinant of the matrix of p and Q is called determining of Sylvester or Résultant from p and Q .

Applications

The equation of Bézout of unknown factors the polynomials X (of degree < m ) and there (of degree < N )

$x\; \backslash \; cdot\; p\; +\; there\; \backslash \; cdot\; Q\; =\; 0$

can be rewritten matriciellement

$S\_\; \{p,\; Q\}\; \backslash \; cdot\; \backslash \; begin\; \{pmatrix\}\; \backslash \; tilde\; X\; \backslash \; \backslash \; \backslash \; tilde\; there\; \backslash \; end\; \{pmatrix\}\; =\; \backslash \; begin\; \{pmatrix\}\; 0\; \backslash \; \backslash \; 0\; \backslash \; end\; \{pmatrix\}$

in which $\backslash \; tilde\; x$ is the vector of size $n$ coefficients of the polynomial X and $\backslash \; tilde\; y$ vector of size $m$.

Thus the core of the matrix of Sylvester gives all the solutions of this equation of Bézout with and .

The row of the matrix of Sylvester is thus connected to the degree of PGCD of $p$ and $q$.

$\backslash \; deg\; (\backslash \; mathrm\; \{pgcd\}\; (p,\; Q))\; =\; m+n-\; \backslash \; mathrm\; \{row\}\; ~S\_\; \{p,\; Q\}$.

In particular, the Résultant from p and Q are null if and only if p and Q has a common factor of degree equal to or higher than one.

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