On the intrinsic complexity of the arithmetic Nullstellensatz

Abstract

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorith mic procedure computing the polynomials and constants occurring in a Bezout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again poly nomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function 7ls(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteris tic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.

Sharp estimates for the arithmetic Nullstellensatz

“…However, technical reasons (see Remark 3.9) prevent us to apply the same principle in this chapter, and we need to carry out a more careful analysis. We note that all aspects of this preparation were previously covered in the research papers [2], [17], [29], [19]. However the bounds presented therein are either non-explicit or not precise enough for our purposes.…”

“…One of its outstanding features is that it performs effective division modulo complete intersection ideals [17], [12], [29], [47], [15], [19]. In this section we apply trace formula to obtain sharp height estimates in the division procedure.…”

“…In particular, they decide the consistency of a given polynomial system. In their arithmetic presentation, they apply to Lojasiewicz inequalities [48], [23] and to the consistency problem over finite fields [25], [19]. We recall that the height h(f ) of a polynomial f ∈ Z Z[x 1 , .…”

“…We refer the reader to the surveys [55], [1], [42] for a broad introduction to the history of the effective Nullstellensatz, main results and open questions. Aside from degree and height estimates, there is a strong current area of research on computational issues [17], [12], [29], [16], [15], [19]. There are other results in the recent research papers [47], [27], [9].…”

“…It was soon realized that the degrees in the Nullstellensatz can be controlled in terms of this parameter, giving rise to the so-called "intrinsic Nullstellensätze" [16], [30], [15], [49]. Recently Hägele, Morais, Pardo and Sombra [19] (see also [18]) obtained an arithmetic analogue of these intrinsic Nullstellensätze. To this aim, they introduced the notion of height of a polynomial system, the arithmetic analogue of the degree of the system.…”