Security guarantees of Shamir's secret sharing when some co-efficients are zero Shamir secret sharing techniques rely on polynomials for splitting and reconstructing. It's security properties are very good, i.e. it is impossible to reconstruct the secret when $t-1$ shares are given. 

The coefficient for the highest term of a polynomial should be non-zero. 
But is it mandatory for other co-efficients of polynomials to be non-zero ? 

Is there an impact to security guarantees when few co-efficients (other than the highest term) are zero ?
The coefficients must be uniformly chosen. If you do not choose your coefficients uniformly, then by Kerckhoff's principle, the attacker knows this, and that makes it easier than normal to reconstruct the polynomial, and thus to obtain the secret.