# Matthew Russell # Experimental Math # Homework 7 # I give permission to post this. ##################################################################################### # 1. Write a procedure Gsomosk(Ini,k,n): that outputs the n-th term of the # Somos-k recurrence, where Somos-k is the natural k-th order analog # of the Somos-4 recurrence (first think what it is). GSomosk:=proc(Ini,k,n) option remember: if n<=nops(Ini) then return Ini[n]: else return (add(GSomosk(Ini,k,n-j)*GSomosk(Ini,k,n-k+j),j=1..trunc(k/2)))/GSomosk(Ini,k,n-k): fi: end: ##################################################################################### # 2. Find out, experimentally, for which k, Somos-k, with Ini=[1\$k], seem to # always produce integers. # Somos-1 does not make sense. # Somos-2 and Somos-3 are the constant all-ones sequence. # Somos-4, Somos-5, Somos-6, and Somos-7 all appear to be integer sequences. # Somos-k, for k>=8, appear to be not be integer sequences. ##################################################################################### # 3. Find out, experimentally, for which k, Somos-k, with Ini=[x[1], ..., x[k]], # seem to always produce Laurent polynomials (rational functions whose denominators are monomials) # Somos-k for k<=7 seems to always produce Laurent polynomials. # Somos-k for k>=8 seems to always not produce Laurent polynomials. ##################################################################################### # 4. The total degree of a monomial x1^a1 x2^a2 x3^a3 ... is a1+a2+a3+... # a homogenous polynomial of degree d is one for which each of its terms # is a monomial of degree d. Adapt GenPol(X,d,a,co), to write a procedure # GenHomPol(X,d,a,co) that yields a homog. polynomial of total degree d. GenHomPol:=proc(X,d,a,counter) local x,m,counter1,i,P1,X1,P,var: m:=nops(X): counter1:=counter: x:=X[1]: if m=1 then return a[counter1]*x^d,{a[counter1]},counter1+1: else P:=0: var:={}: X1:=[op(2..m,X)]: for i from 0 to d do P1:=GenHomPol(X1,i,a,counter1): P:=expand(P+P1[1]*x^(d-i)): var:=var union P1[2]: counter1:=P1[3]: od: return P,var,counter1: fi: end: ##################################################################################### # 5. Using GenHomPol(X,d,a,co), write a program GenNonHomPol(X,d,a,co) of # total degree d (in other words one that is the sum of a homog. polynomials of total degree i for i=0...d) GenNonHomPol:=proc(X,d,a,counter) local i,P,var,counter1,G: P:=0: var:={}: counter1:=counter: for i from 0 to d do G:=GenHomPol(X,i,a,counter1): P:=P+G[1]: var:=var union G[2]: counter1:=G[3]: od: return P,var,counter1: end: #####################################################################################