How many independently-developed writing systems have there been? One hundred years ago, educated people would have told you without hesitation, “three: Sumerian or Egyptian in the Middle East, Chinese in China, and Mayan in Central America”. Very well-educated people would say four, including Indus script.
Unfortunately, archaeologists keep messing up the picture. In addition to finding Vinca and eight Chinese maybe-they-are-writing-maybe-they-aren’t symbols afterwards, in 1986 people fished a big slab of rock out of a stream that with symbols on it that is clearly writing. This is recent enough that the academics haven’t settled on a name, but “Epi-Olmec”, “Isthmian”, and “La Mojarra” script are all names used for this script.
They have also found fragments of script which appears to be Zapotec, which might predate Epi-Olmec by a bit, but that dating is disputed.
Then in the early 1990s, some road builders in an area that had been inhabited by Olmecs stumbled over a smaller stone slab, now called the Cascajal block, with symbols on it. The block has been dated to around 900 BC, and sure looks like writing to me: it has 28 unique symbols in a string that is 62 characters long. The block is rather large and the characters are on a side of the block that has been deliberately flattened/smoothed.
Unfortunately, there aren’t enough characters or an obvious enough relationship to other scripts to be able to decipher the Cascajal glyphs. Thus, Cascajal will have to stay in the maybe-writing-maybe-not category.
Epi-Olmec, on the other hand, is clearly writing, and is clearly related to the other mesoamerican writing systems. In addition to artistic similarities, all of the mesoamerican writing systems use the same notations for numbers and for dates. So regardless of whether or not the Cascajal block was writing or not, Mayan was not the first mesoamerican writing system. Sorry, Mayan.
However, the good news for hemispheral pride is that writing was clearly independently invented in the New World. While one can imagine scenarios where at least the idea of writing travelled from the Middle East to East and South Asia, camel caravans are highly unlikely to have reached Mexico.
Links: Wikipedia, Ancient Scripts. I also highly recommend the Ancient Scripts article on mesoamerican writing systems.