Noether normalization lemma

Abstract: In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map onto the $d$-dimensional affine space. Equivalently, for every integral differential algebra $A$ over differential field of zero characteristic there exist differentially independent $b_1, \ldots, b_d$ such that $A$ is differentially algebraic over subalgebra $B$ differentially generated by $b_1, \ldots, b_d$, and whenever $\mathfrak{p} \subset B$ is a prime differential ideal, there exists a prime differential ideal $\mathfrak{q} \subset A$ such that $\mathfrak{p} = B \cap \mathfrak{q}$. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations.

Abstract: We study the problem of obtaining deterministic black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an deterministic white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work there was no known such black-box algorithm. The main result of this work gives the first quasi-polynomial sized hitting sets for size S circuits from this class, when the order of the variables is known. As our hitting set is of size exp(lg^2 S), this is analogous (in the terminology of boolean pseudorandomness) to a seed-length of lg^2 S, which is the seed length of the pseudorandom generators of Nisan and Impagliazzo-Nisan-Wigderson for read-once oblivious boolean branching programs. Our results are stronger for branching programs of bounded width, where we give a hitting set of size exp(lg^2 S/lglg S), corresponding to a seed length of lg^2 S/lglg S. This is in stark contrast to the known results for read-once oblivious boolean branching programs of bounded width, where no pseudorandom generator (or hitting set) with seed length o(lg^2 S) is known. In follow up work, we strengthened a result of Mulmuley, and showed that derandomizing a particular case of the Noether Normalization Lemma is reducible to black-box PIT of read-once oblivious ABPs. Using the results of the present work, this gives a derandomization of Noether Normalization in that case, which Mulmuley conjectured would difficult due to its relations to problems in algebraic geometry. We also show that several other circuit classes can be black-box reduced to read-once oblivious ABPs, including set-multilinear ABPs (a generalization of depth-3 set-multilinear formulas), non-commutative ABPs (generalizing non-commutative formulas), and (semi-)diagonal depth-4 circuits (as introduced by Saxena).

Abstract: In this paper, we prove the following differential analog of the Noether normalization lemma: for every $d$-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map onto the $d$-dimensional affine space. Equivalently, for every integral differential algebra $A$ over differential field of zero characteristic there exist differentially independent $b_1, \ldots, b_d$ such that $A$ is differentially algebraic over subalgebra $B$ differentially generated by $b_1, \ldots, b_d$, and whenever $\mathfrak{p} \subset B$ is a prime differential ideal, there exists a prime differential ideal $\mathfrak{q} \subset A$ such that $\mathfrak{p} = B \cap \mathfrak{q}$. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations.