Real zeros of mixed random fewnomial systems

Consider a system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $f_i$ has a prescribed set of exponent vectors described by a set $A_i \subseteq \mathbb{Z}^n$ of cardinality $t_i$, whose convex hull is denoted $P_i$. Assuming that the coefficients of the $f_i$ are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most $(2\pi)^{-\frac{n}{2}} V_0 (t_1-1)\ldots (t_n-1)$. Here $V_0$ denotes the number of vertices of the Minkowski sum $P_1+\ldots + P_n$. However, this bound does not improve over the bound in B\"urgisser et al. (SIAM J. Appl. Algebra Geom. 3(4), 2019) for the unmixed case, where all supports $A_i$ are equal. All arguments equally work for real exponent vectors.Comment: 10 pages. Fixed an error in the interpretation of the old Theorem 1.3, which was hence downgraded to Proposition 1.3. Added a reference, put some minor clarifications and fixed some typos. Converted to ACM two column styl

On the Number of Real Zeros of Random Sparse Polynomial Systems

Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $\mathfrak{f}_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size $t_k$. Assuming that the coefficients of the $\mathfrak{f}_k$ are independent Gaussian of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by $4^{-n} \prod_{k=1}^n t_k(t_k-1)$. This result is a probabilisitc version of Kushnirenko's conjecture; it provides a bound that only depends on the number of terms and is independent of their degree.Comment: 26 pages. Different original titl

Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems

Given convex polytopes $P_1,...,P_r$ in $R^n$ and finite subsets $W_I$ of the Minkowsky sums $P_I=\sum_{i \in I} P_i$, we consider the quantity $N(W)=\sum_{I \subset {\bf [}r {\bf ]}} {(-1)}^{r-|I|} \big| W_I \big|$. We develop a technique that we call irrational mixed decomposition which allows us to estimate $N(W)$ under some assumptions on the family $W=(W_I)$. In particular, we are able to show the nonnegativity of $N(W)$ in some important cases. The quantity $N(W)$ associated with the family defined by $W_I=\sum_{i \in I} W_i$ is called discrete mixed volume of $W_1,...,W_r$. We show that for $r=n$ the discrete mixed volume provides an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $W_1,...,W_n$. We also prove that the discrete mixed volume of $W_1,...,W_r$ is bounded from above by the Kouchnirenko number $\prod_{i=1}^r (|W_i|-1)$. For $r=n$ this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $W_1,...,W_n$. This conjecture was disproved, but our result shows that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking.Comment: 27 pages, small corrections in version

Maximally positive polynomial systems supported on circuits

A real polynomial system with support \calW \subset \Z^n is called {\it maximally positive} if all its complex solutions are positive solutions. A support \calW having $n+2$ elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit \calW \subset\Z^n is at most m(\calW)+1, where m(\calW) \leq n is the degeneracy index of \calW. We prove that if a circuit \calW \subset \Z^n supports a maximally positive system with the maximal number m(\calW)+1 of non-degenerate positive solutions, then it is unique up to the obvious action of the group of invertible integer affine transformations of $\Z^n$. In the general case, we prove that any maximally positive system supported on a circuit can be obtained from another one having the maximal number of positive solutions by means of some elementary transformations. As a consequence, we get for each $n$ and up to the above action a finite list of circuits \calW \subset \Z^n which can support maximally positive polynomial systems. We observe that the coefficients of the primitive affine relation of such circuit have absolute value $1$ or $2$ and make a conjecture in the general case for supports of maximally positive systems

Descartes' Rule of Signs for Polynomial Systems supported on Circuits

We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the exponent vectors and the given coefficients. We show that our bound is sharp and is related to the signature of the circuit.Comment: 25 pages, 3 figure

Quantitative Aspects of Sums of Squares and Sparse Polynomial Systems

Computational algebraic geometry is the study of roots of polynomials and polynomial systems. We are familiar with the notion of degree, but there are other ways to consider a polynomial: How many variables does it have? How many terms does it have? Considering the sparsity of a polynomial means we pay special attention to the number of terms. One can sometimes profit greatly by making use of sparsity when doing computations by utilizing tools from linear programming and integer matrix factorization. This thesis investigates several problems from the point of view of sparsity. Consider a system F of n polynomials over n variables, with a total of n + k distinct exponent vectors over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also give a complete classification for when an n-variate n + 2-nomial positive polynomial can be written as a sum of squares of polynomials. Finally, we investigate the problem of approximating roots of polynomials from the viewpoint of sparsity by developing a method of approximating roots for binomial systems that runs more efficiently than other current methods. These results serve as building blocks for proving results for less sparse polynomial systems