Computing the isotopy type of a hypersurface, defined as the positive real zero set of a multivariate polynomial, is a challenging problem in real algebraic geometry. We focus on the case where the defining polynomial has combinatorially restricted exponent vectors and fixed coefficient signs, enabling faster computation of the isotopy type. In particular, Viro's patchworking provides a polyhedral complex that has the same isotopy type as the hypersurface, for certain choices of the coefficients. So we present properties of the signed support, focusing mainly on the case of n-variate (n+3)-nomials, that ensure all possible isotopy types can be obtained via patchworking. To prove this, we study the signed reduced A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions.

Suppose f is a polynomial in n variables with real coefficients and exactly n + k monomial terms. Estimating the number of connected components of the zero set of f in the positive orthant is a fundamental problem in real algebraic geometry, and tight estimates have applications in computational complexity and topology. We prove that, for generic coefficients and exponent vectors, the number of connected components is at most 3 when k = 3, settling an open question from Fewnomial Theory. Our results also extend to exponential sums with real exponents. A key contribution is a deeper analysis of the underlying \(\mathcal {A}\)-discriminant contours, which should be useful for other quantitative geometric problems.

Suppose f is a polynomial in n variables with degree d, exactly n + k monomial terms, coefficients in { ± 1, …, ±H} for some <?TeX $H\!\in \!\mathbb {N}$?> Math 1 , and Newton polytope of positive volume. Testing real feasibility of such an f is a fundamental task whose bit-complexity remains a mystery, even in the first non-trivial case k = 2: The fastest algorithms so far have deterministic bit-complexity (nlog (dH))O(n). We prove a significant speed-up that holds for all but a small collection of inputs in the k = 2 case: Bit complexity (nlog (dH))O(1) for all but a <?TeX $O\!\left(\frac{1}{2^n H}\right)$?> Math 2 -fraction of the f above, for any fixed support. Our result follows by combining a connection to diophantine approximation with a more recent anti-concentration result. In particular, we show that for random inputs, Baker’s famous theorem on linear forms in logarithms can be significantly sharpened. We also consider extensions beyond feasibility such as counting connected components and systems of circuit polynomials.