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Jacques Gélinas on Mon, 23 Jul 2018 18:31:22 +0200
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Using taylor() with polynomials in many variables
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- To: "pari-users@pari.math.u-bordeaux.fr" <pari-users@pari.math.u-bordeaux.fr>
- Subject: Using taylor() with polynomials in many variables
- From: Jacques Gélinas <jacquesg00@hotmail.com>
- Date: Mon, 23 Jul 2018 16:31:18 +0000
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- Thread-topic: 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