41 Use the procedure in Exercise 38 to express Ei x7x as a polynomial in...

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41 Use the procedure in Exercise 38 to express Ei x7x as a polynomial in the elementary symmetric functions We now know that a symmetric polynomial f x xn can be written uniquely as a poly nomial in the elementary symmetric functions Using this existence and uniqueness we can describe an alternate and computationally useful method for determining the coefficients of the elementary symmetric functions in this polynomial As in Exercise 37 we may assume that f x x is homogeneous of degree M Let N be the maximum degree of any of the variables x x in f x Xn a Determine all of the possible monomials from the constraints Ass appearing in f x Xn a1 2a2 a a2 nan M an N b Since f x Xn A 2 s is a polynomial identity it is valid for any substitution of values for x Xn Each substitution into this equation gives a linear relation on the coefficients A and so a sufficient number of substitutions will

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41 Use the procedure in Exercise 38 to express Ei x7x as a polynomial in the elementary symmetric functions We now know that a symmetric polynomial f x xn can be written uniquely as a poly nomial in the elementary symmetric functions Using this existence and uniqueness we can describe an alternate and computationally useful method for determining the coefficients of the elementary symmetric functions in this polynomial As in Exercise 37 we may assume that f x x is homogeneous of degree M Let N be the maximum degree of any of the variables x x in f x Xn a Determine all of the possible monomials from the constraints Ass appearing in f x Xn a1 2a2 a a2 nan M an N b Since f x Xn A 2 s is a polynomial identity it is valid for any substitution of values for x Xn Each substitution into this equation gives a linear relation on the coefficients A and so a sufficient number of substitutions will