When performing a truncated multiplication, the checks for the overflow in the exponents are not needed and actually harmful.
As an example, when performing a substitution in a polynomial (say p = x^14), I may want to substitute x with y^14. This could overflow the exponent type and throw. But in the ring of truncated polinomials such a substitution is perfectly valid and can be represented as the resulting poly would not contain terms overflowed as they are truncated. This is an important issue, for example, in the software audi/pyaudi when inverting Taylor maps.
When performing a truncated multiplication, the checks for the overflow in the exponents are not needed and actually harmful.
As an example, when performing a substitution in a polynomial (say p = x^14), I may want to substitute x with y^14. This could overflow the exponent type and throw. But in the ring of truncated polinomials such a substitution is perfectly valid and can be represented as the resulting poly would not contain terms overflowed as they are truncated. This is an important issue, for example, in the software audi/pyaudi when inverting Taylor maps.