Jacques Gélinas on Mon, 23 Jul 2018 18:31:22 +0200

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Using taylor() with polynomials in many variables

Consider the polynomial expression

E = r*(n-1)*(2*r+1)*(2+xx)-3*(n-r)*(2*r+1-x*y);

and I want to show it is positive under the constraints
n=2,3,4,... ;  r=2,...,n-1;  1<= x,y <= 1;

Since E>0 should hold for all integers n, the coefficients of the
powers of n-2 could all be positive under the other constraints.

====>  How can PARI/GP give me these taylor coefficients ???    <=====

taylor(subst(E,n,m+2),m) ==
(-3*y*r + (3*m + 6)*y)*x + (((2*m + 2)*xx + (4*m + 10))*r^2 + ((m + 1)*xx +
 (-4*m - 7))*r + (-6 - 3*m + O(m^16)))

I **think** the taylor series in m is found, but cannot be shown because of
the variables display order (x and y have precedence).
And of course, I do not need the O() term.

Jacques Gélinas