Let $F: C^n rightarrow C^m$ be a polynomial map with $degF=d geq 2$. We prove that $F$ is invertible if $m = n$ and $sum^{d-1}_{i=1} JF(alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = {beta + mu gamma | mu in C} subseteq C^n$ ($gamma e 0$), $F|_L$ is linearly rectifiable, if and only if $sum^{d-1}_{i=1} JF(alpha_i) cdot gamma e 0$ for all $alpha_i in L$. This appears to be the case for all affine lines $L$ when $F$ is injective and $d le 3$. We also prove that if $m = n$ and $sum^{n}_{i=1} JF(alpha_i)$ is invertible for all $alpha_i in C^n$, then $F$ is a composition of an invertible linear map and an invertible polynomial map $X+H$ with linear part $X$, such that the subspace generated by ${JH(alpha) | alpha in C^n}$ consists of nilpotent matrices.
The ring R of real-exponent polynomials in n variables over any field has global dimension n+1 and flat dimension n. In particular, the residue field k = R/m of R modulo its maximal graded ideal m has flat dimension n via a Koszul-like resolution. Projective and flat resolutions of all R-modules are constructed from this resolution of k. The same results hold when R is replaced by the monoid algebra for the positive cone of any subgroup of $mathbb{R}^n$ satisfying a mild density condition.
We give three determinantal expressions for the Hilbert series as well as the Hilbert function of a Pfaffian ring, and a closed form product formula for its multiplicity. An appendix outlining some basic facts about degeneracy loci and applications to multiplicity formulae for Pfaffian rings is also included.
Let $R=mathbf{C}[xi_1,xi_2,ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $mathfrak{S}$. We classify the $mathfrak{S}$-primes of $R$, determine the containments among these ideals, and describe the equivariant spectrum of $R$. We emphasize that $mathfrak{S}$-prime ideals need not be radical, which is a primary source of difficulty. Our results yield a classification of $mathfrak{S}$-ideals of $R$ up to copotency. Our work is motivated by the interest and applications of $mathfrak{S}$-ideals seen in recent years.