symmetric systems of polynomial equations

Suppose I have a polynomial p(x1,…,xN) in N complex variables, and I wish to solve p(xπ(1),…,xπ(N))=0 for all permutations π∈SN. Clearly this is overdetermined for generic p, but suppose p is symmetric under exchange of all but one variable. Then this gives N distinct equations, and so generically one expects a discrete set of solutions. … Read more

Zeros of polynomials modulo a non-prime

Suppose I have a set S and I want to find a polynomial p such that p(s)=0modn if s∈S, and that it is non-zero modulo n otherwise. In the literature such an S is sometimes called a root set (see Finding such a polynomial is always possible if n is a prime number, but … Read more

numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for e−x, because a computation directly from the polynomial coefficients has poor numerical stability. I finally blundered upon an approach alluded to a paper by Campos and Calderón and a related item mentioned in a paper by … Read more

An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent transpositions, any permutation $w\in S_n$ can be written as $w=s_{i_1}s_{i_2}\cdots s_{i_l}$ and an expression $w=s_{i_1}s_{i_2}\cdots s_{i_l}$ of minimal possible length $l$ is called a reduced decomposition, $l=\ell(w)$ … Read more

Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads n∑k=0Hk(x)Hk(y)2kk!=12n+1n!Hn+1(x)Hn(y)−Hn(x)Hn+1(y)x−y. I am interested in a similar identity, where one of the indices is shifted by one, explicitly Fn(x,y):=n∑k=0Hk+1(x)Hk(y)2k+1(k+1)!=xn∑k=01k+1Hk(x)Hk(y)2kk!−n∑k=0kk+1Hk−1(x)Hk(y)2kk!, where the recursion relation Hk+1(x)=2xHk(x)−2kHk−1(x) was used. In the first term Christoffel-Darboux cannot be applied due to the 1/(k+1) prefactor and the second term is not quite Fn−1(y,x) … Read more

When a ring is a polynomial ring?

In the paper (2.11) the authors show that if k∗ is a separable algebraic extension of k and x1,x2,…,xn are indeterminates over k∗ and a normal one dimensional ring A with k⊂A⊂k∗[x1,x2,…,xn] then A has the form k′[t] where k′ is the algebraic closer of k in A. The above is a very strict sufficient … Read more

Spectrum of Kernel – Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form K(m,n)=e−β4(m+n+1)22+m+n2√m!n!√πn−m[1Γ(−m/2)Γ(−n+12)−1Γ(−n/2)Γ(−m+12)] where m,n∈N and 0<q<1 a real parameter. I want to diagonalize this kernel and find its eigenvalues/eigenfunctions. It looks like an integrable kernel after having performed the Cristoffel-Darboux summation formula which has a general form like Kk(n,m)=√w(m)w(n)fk(n)fk−1(m)−fk(m)fk−1(n)n−m. and one should take the … Read more

Difference between Chebyshev first and second degree iterative methods

Consider linear equation Au=f. We want to solve it with iterative method (assuming A is good). First order iterative method is: uk+1=uk−αk+1(Auk−f), The second degree method is: uk+1=uk−αk+1(Auk−f)−β(uk−uk−1). For both methods we can define iteration parameters αk, βk via minimax problem which solution is Chebyshev polynomials. This is good, but it seems to me, that … Read more

Does a polynomial system with precisely e solutions have a Groebner basis of degree bounded by e?

Let k be a field and let R=k[X1,…,Xn] be a polynomial ring. Let F⊂R be a finite subset generating a radical ideal I with precisely e solutions over an algebraic closure of k. Is there a monomial order on R with a Groebner basis of degree ≤e for I? Answer AttributionSource : Link , Question … Read more

When is a given polynomial a square of another polynomial?

I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let’s consider the polynomial f(x):=1−x+2bxn−2bxn+1−b2x2n−1+2b2x2n−b2x2n+1−2bx3n−1+2bx3n−x4n−1+x4n∈k[x], where k is a field of characteristic p>0, n>2 is an integer, and b∈k with b≠0,1,−1. Indeed, in the context I meet, the field k is just the … Read more