The Resolvents of Knig And Other Types of Symmetric Functions
The Resolvents of Knig And Other Types of Symmetric Functions
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^, . , .
01 F, -rr, wjr-o, ' ' • >.^, _, . Smce these expressions are linear in X and do not vanish identically for every value of x, then the identical vanishing of each of these r expressions necessitates the vanishing of two seminvariants, and hence the simultaneous vanishing of 2r seminvariants furnishes the minimum set of necessary and sufficient conditions that ax"" be divisible by the rth power of ^J^. In particular, if ax"* contain ^J^ as a factor, then from (4) and (5) j;(m-l/2) t i+l ...(7) '- ^ ' ^(m-l/2) t i f=0 [^ and conversely. These two conditions are equivalent to the redundant set of m + 1 conditions furnished by the identical vanishing of the co variant $, given by Clebsch, * which for w = 4 is $ = - 2D{aO^, 'ax' + 2^x'{a^')ian^Jax = 0, where D is the discriminant of ^^^ and ^^ = ^J"^ = ^x"^- B. Igelf has developed a method depending upon the vanishing of a series of resultants of every two forms of a certain system.
* Theorie der Binaren Formen, p. 94.
■\ Sitzungsherichte der Kon.
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To quickly assess the difficulty of the text, read a short excerpt:
^, . , .
01 F, -rr, wjr-o, ' ' • >.^, _, . Smce these expressions are linear in X and do not vanish identically for every value of x, then the identical vanishing of each of these r expressions necessitates the vanishing of two seminvariants, and hence the simultaneous vanishing of 2r seminvariants furnishes the minimum set of necessary and sufficient conditions that ax"" be divisible by the rth power of ^J^. In particular, if ax"* contain ^J^ as a factor, then from (4) and (5) j;(m-l/2) t i+l ...(7) '- ^ ' ^(m-l/2) t i f=0 [^ and conversely. These two conditions are equivalent to the redundant set of m + 1 conditions furnished by the identical vanishing of the co variant $, given by Clebsch, * which for w = 4 is $ = - 2D{aO^, 'ax' + 2^x'{a^')ian^Jax = 0, where D is the discriminant of ^^^ and ^^ = ^J"^ = ^x"^- B. Igelf has developed a method depending upon the vanishing of a series of resultants of every two forms of a certain system.
* Theorie der Binaren Formen, p. 94.
■\ Sitzungsherichte der Kon.
What to read after The Resolvents of Knig And Other Types of Symmetric Functions?
You can find similar books in the "Read Also" column, or choose other free books by Stanley Pulliam Shugert to read onlineMoreLess
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