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If u0,u1,...,u4 are variables defined in R=PolynomialRing(QQ,5,'u') and RR.<T>=R[] is the ring where I defined a polynomial (T-u0)*(T-u1)*...*(T-u4), I would like to compute the elementary symmetrical functions on these variables: namely something like e[1,1](u0,...u4) to get u0u1+u0u2+...+u3u4.

In the documentation about S=SymmetricFunctions(QQ);e=S.monomials(), I don't see how you can assign anything.

Also, but this is a secondary problem, this space S is a vector space, generated by a huge number of functions. Is it possible to get it generated as an algebra by the much fewer elementary symmetrical functions, the e[1,1,...,1]s?

If u0,u1,...,u4 are variables defined in R=PolynomialRing(QQ,5,'u') and RR.<T>=R[] is the ring where I defined a polynomial (T-u0)*(T-u1)*...*(T-u4), I would like to compute the elementary symmetrical functions on these variables: namely something like e[1,1](u0,...u4) to get u0u1+u0u2+...+u3u4.

In the documentation about S=SymmetricFunctions(QQ);e=S.monomials(), I don't see how you can assign anything.

Also, but this is a secondary problem, this space rename variables by "alphabet", eg Sv=e[1,1,1].expand(4,alphabet='u') and then coerce this in the ring, is R(v) but if I apply a vector space, generated permutation, say p=SymmetricGroup(4).subgroup([(1,2)]) to it, by a huge number of functions. Is it possible to get it generated as an algebra by the much fewer elementary symmetrical functions, the e[1,1,...,1]R(v)(*p[1](R.gens()))s?, nothing happens.

If u0,u1,...,u4 are variables defined in R=PolynomialRing(QQ,5,'u') and RR.<T>=R[] is the ring where I defined a polynomial (T-u0)*(T-u1)*...*(T-u4), I would like to compute the elementary symmetrical functions on these variables: namely something like e[1,1](u0,...u4) to get u0u1+u0u2+...+u3u4.

I thnik the following works. In the documentation about S=SymmetricFunctions(QQ);e=S.monomials(), you can rename variables by "alphabet", eg v=e[1,1,1].expand(4,alphabet='u') and then coerce this in the ring, R(v) but if I . Then one may apply a permutation, say p=SymmetricGroup(4).subgroup([(1,2)]) to it, by R(v)(*p[1](R.gens())), nothing happens.which seems to work!

If u0,u1,...,u4 are variables defined in R=PolynomialRing(QQ,5,'u') and RR.<T>=R[] is the ring where I defined a polynomial (T-u0)*(T-u1)*...*(T-u4), I would like to one can compute the elementary symmetrical functions on these variables: namely something like e[1,1](u0,...u4) to get u0u1+u0u2+...+u3u4.

I thnik the following works. In the documentation about S=SymmetricFunctions(QQ);e=S.monomials(), you can rename variables by "alphabet", eg v=e[1,1,1].expand(4,alphabet='u') and then coerce this in the ring, R(v). Then To check, one may apply a permutation, say p=SymmetricGroup(4).subgroup([(1,2)]) to it, by R(v)(*p[1](R.gens())), which seems to work!

If u0,u1,...,u4 are variables defined in R=PolynomialRing(QQ,5,'u') and RR.<T>=R[] is the ring where I defined a polynomial (T-u0)*(T-u1)*...*(T-u4), one can How to compute the elementary symmetrical elementart symmetric functions on these variables: namely something like e[1,1](u0,...u4) to get u0u1+u0u2+...+u3u4.

In the documentation about S=SymmetricFunctions(QQ);e=S.monomials(), you can rename variables by "alphabet", eg v=e[1,1,1].expand(4,alphabet='u') and then coerce this in the ring, R(v). To check, one may apply a permutation, say p=SymmetricGroup(4).subgroup([(1,2)]) to it, by R(v)(*p[1](R.gens())), which seems to work!

Now, if the uie[1,1,..]'s are not to calculate variables but calculated ones, say a,b,c,... then how to assign the e[1,1,..] to calculate teh elementary symmetrical functions , typically computing "elemenfs" on these a,b,c,...?roots of a polynomial? This seems such a basic question, but I can't find anywhere.

How to compute the elementart elementary symmetric functions e[1,1,..]'s to calculate of given variables a,b,c,..., typically computing "elemenfs" on roots of a polynomial? This seems such a basic question, but I can't find anywhere.

How to compute the elementary symmetric functions e[1,1,..]'s of given variables a,b,c,..., typically computing "elemenfs" "elements" on roots of a polynomial? This seems such a basic question, but I can't find anywhere.