Suppose \(L\) is a finite extension of the field of \(p\)-adic numbers, and \(k\) is a positive integer. The author proves an upper bound for the positive integer \(B=B(k,L)\) with the following property: if \(f\), \(f\neq 0\), is a polynomial with coefficients in \(L\), and has at most \(k+1\) non-zero terms, then \(f\) has at most \(B\) nonvanishing zeros in \(L\) counted with multiplicities. For trinomials, \(f=a+bx^t+cx^u\), where \(a, b, c\) are \(2\)-adic numbers, the bound \(B\leq 6\) is shown to be sharp.
As a consequence, the author obtains an upper bound for the degree \(A=A(m,k,d)\) of \(g\), a divisor of \(f\), such that every irreducible factor is of degree at most \(d\). Here, \(f\) is a non-zero polynomial with coefficients in a number field of degree less than or equal to \(m\), and with at most \(k+1\) non-zero terms. In particular, by Descartes’ rule of signs, one can take \(A(1,k,1)=2k\). The proofs are based on certain non-archimedean estimates. The author asks for the best values of the involved constants.
For the entire collection see [Zbl 0911.00025].