Algebraic Invariants
Algebraic Invariants
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To quickly assess the difficulty of the text, read a short excerpt:
, ap.
Let W be the weight of K and hence of the coefficient of y. Then k is of total degree W in the a's and of degree co in x/y. Thus K = 2C(| product of w differences hke — aA [ y J •{product of W — o) differences like ar—as].
Hence K = ao'^ZCi\'prod\iQt of w differences like x—ary] • {product of W—oi differences like ar—as\.
Next, for x=—t], y=^, f becomes F = Ao^^+ . . . with a root — X/ut corresponding to each root ar of /. The function K for F is product of CO differences like ^-\ — ri = ...- !^[ ar —ar \ • ] product of W — o) differences like — ^ [ .
\ aras J Using the value of Aq in § 35, we see that the factor \-\yW . . . ap'^ must be cancelled by the —ar and the aras in the denominators.
5S ALGEBEAIC INVARIANTS Thus each term of the sum involves every root exactly d times.
The signs agree since as follows by counting the total number of as.
Any covariant of degree d, order co and weight W of ao{x—aiy) . . . (x—apy) equals the product of ao^ by a sum of products of constants and w differences like x—ary and W — oj differences like ar—as, such that every root occurs in exactly d factors of each product; more- over, the sum equals a symmetric function of the roots.
What to read after Algebraic Invariants?
You can find similar books in the "Read Also" column, or choose other free books by Leonard E Leonard Eugene Dickson to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
, ap.
Let W be the weight of K and hence of the coefficient of y. Then k is of total degree W in the a's and of degree co in x/y. Thus K = 2C(| product of w differences hke — aA [ y J •{product of W — o) differences like ar—as].
Hence K = ao'^ZCi\'prod\iQt of w differences like x—ary] • {product of W—oi differences like ar—as\.
Next, for x=—t], y=^, f becomes F = Ao^^+ . . . with a root — X/ut corresponding to each root ar of /. The function K for F is product of CO differences like ^-\ — ri = ...- !^[ ar —ar \ • ] product of W — o) differences like — ^ [ .
\ aras J Using the value of Aq in § 35, we see that the factor \-\yW . . . ap'^ must be cancelled by the —ar and the aras in the denominators.
5S ALGEBEAIC INVARIANTS Thus each term of the sum involves every root exactly d times.
The signs agree since as follows by counting the total number of as.
Any covariant of degree d, order co and weight W of ao{x—aiy) . . . (x—apy) equals the product of ao^ by a sum of products of constants and w differences like x—ary and W — oj differences like ar—as, such that every root occurs in exactly d factors of each product; more- over, the sum equals a symmetric function of the roots.
What to read after Algebraic Invariants?
You can find similar books in the "Read Also" column, or choose other free books by Leonard E Leonard Eugene Dickson to read onlineMoreLess
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