New Years Eve Matrix

In algebra , the New Year's Eve matrix for two polynomials is a special matrix with the coefficients of the polynomials , the determinant of which gives the resultant of the polynomials. It is named after the British mathematician James J. Sylvester .

definition

Be a commutative ring . For two polynomials and from the polynomial ring with ${\displaystyle R}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle R[X]}$

${\displaystyle f=\sum _{i=0}^{m}f_{i}X^{i}}$ and ${\displaystyle g=\sum _{i=0}^{n}g_{i}X^{i}}$

of degree is called the square matrix ${\displaystyle m,n\geq 1}$ ${\displaystyle (m+n)}$

${\displaystyle \operatorname {Syl} (f,g)={\begin{pmatrix}f_{m}&&\cdots &&f_{0}&&&\\&f_{m}&&\cdots &&f_{0}&&\\&&\ddots &&&&\ddots &\\&&&f_{m}&&\cdots &&f_{0}\\g_{n}&&\cdots &&g_{0}&&&\\&g_{n}&&\cdots &&g_{0}&&\\&&\ddots &&&&\ddots &\\&&&g_{n}&&\cdots &&g_{0}\\\end{pmatrix}}}$

the New Year's Eve matrix to and . In the illustration, unspecified coefficients are to be understood as zero. ${\displaystyle f}$ ${\displaystyle g}$

properties

For is the matrix that emerges from the New Year's Eve matrix by deleting the last rows of coefficients, the last rows of coefficients and the last columns with the exception of the -th The polynomial ${\displaystyle 0\leq i\leq j\leq m+n}$ ${\displaystyle M_{ji}}$ ${\displaystyle j}$ ${\displaystyle f}$ ${\displaystyle j}$ ${\displaystyle g}$ ${\displaystyle 2j+1}$ ${\displaystyle (m+n-i-j)}$

${\displaystyle S_{j}(f,g)=\sum _{i=0}^{j}\left(\det M_{ji}\right)\,X^{i}}$

is then the -th sub-resultant of and ; their guide coefficient ${\displaystyle j}$ ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle \operatorname {psc} _{j}(f,g)=\det M_{jj}}$

is the -th major sub-result coefficient . The -th major sub-result coefficient ${\displaystyle j}$ ${\displaystyle 0}$

${\displaystyle \operatorname {res} (f,g)=\det \operatorname {Syl} (f,g)}$

finally is the resultant of and . ${\displaystyle f}$ ${\displaystyle g}$

meaning

The main sub-result coefficients have an important meaning as a “measure” of the greatest common divisor of polynomials: The degree of for two polynomials not equal to 0 over a commutative factorial integrity ring is exactly the smallest with . ${\displaystyle \operatorname {ggT} (f,g)}$ ${\displaystyle k\geq 0}$ ${\displaystyle \operatorname {psc} _{k}(f,g)\neq 0}$