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08/16/2022
Sturm's Theorem with Endpoints
Sturm's Theorem is a fundamental 19th century result relating the number...
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02/12/2022
Trinomials and Deterministic Complexity Limits for Real Solving
Consider a univariate polynomial f in Z[x] with degree d, exactly t mono...
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07/19/2021
Root Repulsion and Faster Solving for Very Sparse Polynomials Over p-adic Fields
For any fixed field K∈{ℚ_2,ℚ_3,ℚ_5, …}, we prove that all polynomials f∈...
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02/02/2021
Sub-Linear Point Counting for Variable Separated Curves over Prime Power Rings
Let k,p∈ℕ with p prime and let f∈ℤ[x_1,x_2] be a bivariate polynomial wi...
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12/09/2020
Counting Real Roots in Polynomial-Time for Systems Supported on Circuits
Suppose A={a_1,…,a_n+2}⊂ℤ^n has cardinality n+2, with all the coordinate...
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07/26/2020
Computing zeta functions of large polynomial systems over finite fields
In this paper, we improve the algorithms of Lauder-Wan <cit.> and Harvey...
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02/29/2020
A complexity chasm for solving univariate sparse polynomial equations over p-adic fields
We reveal a complexity chasm, separating the trinomial and tetranomial c...
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01/28/2019
A Faster Solution to Smale's 17th Problem I: Real Binomial Systems
Suppose F:=(f_1,…,f_n) is a system of random n-variate polynomials with ...
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09/10/2018
Probabilistic Condition Number Estimates for Real Polynomial Systems II: Structure and Smoothed Analysis
We consider the sensitivity of real zeros of polynomial systems with res...
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08/30/2018
Randomized Polynomial-Time Root Counting in Prime Power Rings
Suppose k,p∈N with p prime and f∈Z[x] is a univariate polynomial with de...
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03/12/2018
Dedekind Zeta Zeroes and Faster Complex Dimension Computation
Thanks to earlier work of Koiran, it is known that the truth of the Gene...
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11/03/2017
Counting Roots of Polynomials Over Prime Power Rings
Suppose p is a prime, t is a positive integer, and f∈Z[x] is a univariat...
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10/02/2017
New Subexponential Fewnomial Hypersurface Bounds
Suppose c_1,...,c_n+k are real numbers, {a_1,...,a_n+k}⊂R^n is a set of ...
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12/11/2016