Improved construction for universality of determinant and permanent

Valiant [L. Valiant, Completeness classes in algebra, in: Proc. 11th Annual ACM Symposium on the Theory of Computing, Atlanta, GA, 1979, pp. 249–261] proved that every polynomial of formula size e is a projection of the (e+2)×(e+2) determinant polynomial. We improve “e+2” to “e+1”, also for a definition of formula size that does not count multiplications by constants as gates. Our proof imitates the “2e+2” proof of von zur Gathen [J. von zur Gathen, Feasible arithmetic computations: Valiant's hypothesis, Journal of Symbolic Computation 4 (1987) 137–172], but uses different invariants and a tighter set of base cases.