When does a Polynomial Ideal Contain a Positive Polynomial?

We use Gröbner bases and a theorem of Handelman to show that an ideal I of R[x1, . . . , xk] contains a polynomial with positive coefficients if and only if no initial ideal inv(I), v ∈ R, has a positive zero. Let R = R[x1, . . . , xk], R = R[x1, . . . , xk] and, considering Laurent polynomials, let R̃ = R[x1 , . . . , xk ], R̃ = R[x ± 1 , . . . , x ± k ]. For a = (a1, . . . , ak) ∈ Z, write x = x1 1 · · · xk k and denote the coefficient of x in p ∈ R̃ by pa. Then p = ∑ a∈Zk pax a and the Newton polytope N(p) of p is the convex hull of the finite set Log(p) = {a ∈ Z : pa 6= 0}. For v ∈ R, let inv(p) be the sum of pax over those a ∈ Log(p) for which the dot product a · v is maximal. For an ideal I ⊂ R and v ∈ R we have the initial ideal inv(I) = 〈inv(p) : p ∈ I〉 ⊂ R and the corresponding variety V(inv(I)) = {z ∈ C : inv(p)(z) = 0 ∀ p ∈ I}. Observe that in the case v = 0 the ideal inv(I) equals I. We write R for the positive reals. Theorem. An ideal I of R contains a nonzero element of R if and only if (R) ∩ V(inv(I)) = ∅ for all v ∈ R. It will be clear that there are analogous statements for ideals of R̃, as well as for ideals of polynomial (or Laurent polynomial) rings over Q or Z instead of R. The question “When does a submodule M of R contain an element of (R \{0})n ?” will be answered in a longer sequel. The present paper, in dealing with the simpler case n = 1, highlights the utility of Gröbner bases in positivity problems. One ingredient of our proof will be the following theorem of Handelman which deals with the case of a principal ideal. Handelman’s Theorem [H]. For p ∈ R the following are equivalent. (a) There exists q ∈ R such that qp ∈ R \ {0}. (b) We have inv(p)(z) 6= 0 for every v ∈ R and z ∈ (R). A short and self-contained account of the proof of Handelman’s theorem may be found in [DT]. The other ingredient we need is the basic theory of Gröbner bases. Everything we use from this theory can be found in the first 50 pages of [AL]. Date: June 15, 2000. 1991 Mathematics Subject Classification. 13P10,26C05.